Answer:
(a)y(t)=30exp(-0.0231t)
(b)y(20)=18.9mg
Step-by-step explanation:
At a particular time t, the mass of a radioactive substance like Cesium-137 is governed by the equation:
N=N₀e⁻ᵏᵗ where k=ln 2/half life
(a)Mass that remains after t years
Half Life= 30 years
k= ln2/30=0.0231
Initial Mass, N₀=30mg
Therefore the mass N that remains at time t
N=N₀e⁻ᵏᵗ
N=30exp(-0.0231t)
y(t)=30exp(-0.0231t)
(b)We want to determine how much of the sample remains after 20 years.
At t=20 years
y(t)=30exp(-0.0231t)
y(20)=30exp(-0.0231X20)
=30 X 0.63
y(20)=18.9mg
Final answer:
23.81 mg of the original 30 mg sample remaining, rounded to two decimal places.
Explanation:
The half-life of a radioactive isotope like cesium-137 is time it takes for half of the original amount of the substance to decay. For cesium-137, this period is 30 years. To find out how much of a sample remains after a specific amount of time, such as 20 years in the given question, we use the decay formula.
Step 1: Let y(t) be the mass remaining after t years.
Step 2: The decay formula is y(t) = y(0) · (1/2)^(t/half-life), where y(0) is the initial mass and t is the time in years.
For cesium-137 after 20 years: y(20) = 30 mg · (1/2)^(20/30).
Calculating the remaining mass: y(20) = 30 mg · (1/2)^(2/3) ≈ 30 mg · 0.7937 ≈ 23.81 mg.
What point is between 4,16 and 16,16
The point between (4,16) and (16,16) is (10,16) as calculated using the midpoint formula. This point is exactly halfway between the given points.
To determine a point between the two points (4,16) and (16,16), we need to calculate the midpoint.
The formula for finding the midpoint M between two points (x1, y1) and (x2, y2) is:[tex]M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)[/tex]
Putting the coordinates (4,16) and (16,16) into the midpoint formula:[tex]M = \left(\frac{4 + 16}{2}, \frac{16 + 16}{2}\right) = (10, 16)[/tex]
Therefore, the point that lies between (4,16) and (16,16) is (10,16).
To better understand the financial burden students are faced with each term, the statistics department would like to know how much their ST201 students are spending on school materials on average. Let’s use our class data to calculate a 95% confidence interval to estimate the average amount ST201 students spend on materials each term. The average from our student survey is $248 and the number of students sampled is 90. Assume . State the question of interest. On average, how much do ST201 students spend on school materials each term? a. (1 point) Identify the parameter. b. Check the conditions. a. (2 points) Does the data come from a random sample? What are some potential biases about the way the data was collected? (1 point) Is the sample size large enough for distribution of the sample mean to be normal according to the rules for Central Limit Theorem?
Answer:
Answer:
a).
The amount spent on school materials for each term of all ST201students
b).
a).
It is not a random sample. This looks like a convenience sampling and there is sampling bias. This sample is not representative of the entire population. Since it is not a random sample it is not appropriate to generalize the results to all students.
b).
The sample size is 80 which is greater than 30. It is large enough to assume normal distribution according to central limit theorem.
c).
mean: $617
z critical value at 95%: 1.96
standard error = σ/sqrt(n) =500/sqrt(80) = 55.9017
lower limit= mean-1.96*se = 617-1.96*55.9017=507.43
upper limit= mean+1.96*se = 617+1.96*55.9017=726.57
d).
The amount spent on school materials for each term for the 80 ST201students is $617. We are 95% confident that amount spent on school materials for each term of all ST201students falls in the interval ($507.43, $726.57).
Step-by-step explanation:
A 30% solution of fertilizer is to be mixed with a 70% solution of fertilizer in order to get 40 gallons of a 60% solution. How many gallons of the 30% solution and 70%
solution should be mixed?
lion of the 30% solution should be mixed?
Answer:10 gallons of 30% solution and 30 gallons of 70% solution should be mixed.
Step-by-step explanation:
Let x represent the number of gallons of the 30% solution that should be mixed.
Let y represent the number of gallons of the 70% solution that should be mixed.
The total number of gallons of the mixture to be made is 40. This means that
x + y = 40
The 30% solution of fertilizer is to be mixed with a 70% solution of fertilizer in order to get 40 gallons of a 60% solution. This means that
0.3x + 0.7y = 0.6 × 40
0.3x + 0.7y = 24- - - - - - - - - - - - - -1
Substituting x = 40 - y into equation 1, it becomes
0.3(40 - y) + 0.7y = 24
12 - 0.3y + 0.7y = 24
- 0.3y + 0.7y = 24 - 12
0.4y = 12
y = 12/0.4
y = 30
x = 40 - y = 40 - 30
x = 10
Discrete or Continuous? Identify the random variables in Exercises 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 as either discrete or continuous. Total number of points scored in a football game
Answer:
Discrete variable
Step-by-step explanation:
We are given the following in the question:
Variable:
Total number of points scored in a football game
Discrete and continuous data:
Discrete data is the data that can be expressed in whole numbers. They cannot take all the values within an interval.Discrete variables are usually counted and not measured.Continuous variable can be expressed in fractions and can take any value within an interval.Continuous variable are usually measured and not counted.Since, total number of points score in a foot game are expressed in whole numbers and cannot be expressed in decimals, they are discrete variable. They cannot take all the values within an interval and they are usually counted.
Thus,
Total number of points scored in a football game is a discrete variable.
Suppose the null hypothesis, H0, is: a sporting goods store claims that at least 70% of its customers do not shop at any other sporting goods stores. What is the Type I error in this scenario? a. The sporting goods store thinks that less than 70% of its customers do not shop at any other sporting goods stores when, in fact, less than 70% of its customers do not shop at any other sporting goods stores. b. The sporting goods store thinks that at least 70% of its customers do not shop at any other sporting goods stores when, in fact, at least 70% of its customers do not shop at any other sporting goods stores. c. The sporting goods store thinks that less than 70% of its customers do not shop at any other sporting goods stores when, in fact, at least 70% of its customers do not shop at any other sporting goods stores. d. The sporting goods store thinks that at least 70% of its customers do not shop at any other sporting goods stores when, in fact, less than 70% of its customers do not shop at any other sporting goods stores.
Answer:
Null hypothesis: [tex]p \geq 0.7[/tex]
Alternative hypothesis: [tex]p<0.7[/tex]
A type of error I for this case would be reject the null hypothesis that the population proportion is greater or equal than 0.7 when actually is not true.
So the correct option for this case would be:
c. The sporting goods store thinks that less than 70% of its customers do not shop at any other sporting goods stores when, in fact, at least 70% of its customers do not shop at any other sporting goods stores.
Step-by-step explanation:
Previous concepts
A hypothesis is defined as "a speculation or theory based on insufficient evidence that lends itself to further testing and experimentation. With further testing, a hypothesis can usually be proven true or false".
The null hypothesis is defined as "a hypothesis that says there is no statistical significance between the two variables in the hypothesis. It is the hypothesis that the researcher is trying to disprove".
The alternative hypothesis is "just the inverse, or opposite, of the null hypothesis. It is the hypothesis that researcher is trying to prove".
Type I error, also known as a “false positive” is the error of rejecting a null hypothesis when it is actually true. Can be interpreted as the error of no reject an alternative hypothesis when the results can be attributed not to the reality.
Type II error, also known as a "false negative" is the error of not rejecting a null hypothesis when the alternative hypothesis is the true. Can be interpreted as the error of failing to accept an alternative hypothesis when we don't have enough statistical power.
Solution to the problem
On this case we want to test if the sporting goods store claims that at least 70^ of its customers, so the system of hypothesis would be:
Null hypothesis: [tex]p \geq 0.7[/tex]
Alternative hypothesis: [tex]p<0.7[/tex]
A type of error I for this case would be reject the null hypothesis that the population proportion is greater or equal than 0.7 when actually is not true.
So the correct option for this case would be:
c. The sporting goods store thinks that less than 70% of its customers do not shop at any other sporting goods stores when, in fact, at least 70% of its customers do not shop at any other sporting goods stores.
A surfboard shaper has to limit the cost of development and production to $288 per surfboard. He has already spent $61,466.00 on equipment for the boards. The development and production costs are $142 per board. The cost per board is 142x /x+ 61,466 /x dollars. Determine the number of boards that must be sold to limit the final cost per board to $ 288.
How many boards must be sold to limit the cost per board to$288?
Answer:
At least 421 units of boards need to be sold to limit the cost per board to $288
Step-by-step explanation:
Let the number of surfboards made or sold be x
Total cost = fixed cost + variable cost
Fixed Cost = $61466
Variable Cost = 142 × x = $142x
Total cost = 61466 + 142x
Revenue = unit price × quantity = 288×x = 288x
The number of boards that needs to be sold to limit the cost off a board to $288 is the number of units at the point where the total cost matches the revenue.
61466 + 142x = 288x
288x - 142x = 61466
146x = 61466
x = 421 units.
The data below are the number of absences and the final grades of 9 randomly selected students from a literature class. Find the equation of the regression line for the given data.What would be the predicted final grade if a student was absent 14 times? Round the regression line values to the nearest hundredth. Round the predicted grade to the nearest whole number Number of absences X 0,3,6, 4,9,2, 15,8,5 Final grade Y 98,86, 80,82, 71,92, 55,76,82
Answer:
The regression equation is:
Final Grade = 96.14 - 2.76 Number of absence
A student who was absent for 14 days received a final grade of 58.
Step-by-step explanation:
The general form a regression equation is:
[tex]y=\alpha +\beta x[/tex]
Here,
y = dependent variable = Final grade
x = independent variable = Number of absence
α = intercept
β = slope
The formula to compute the intercept and slope are:
[tex]\alpha =\frac{\sum Y. \sum X^{2}-\sum X.\sum XY}{n.\sum X^{2}-(\sum X)^{2}}[/tex]
[tex]\beta =\frac{n.\sum XY-\sum X.\sum Y}{n.\sum X^{2}-(\sum X)^{2}}[/tex]
The value of α and β are computed as follows:
[tex]\alpha =\frac{\sum Y. \sum X^{2}-\sum X.\sum XY}{n.\sum X^{2}-(\sum X)^{2}}=\frac{(722\times460-(52\times3732)}{(9\times460)-(52)^{2}} =96.139\approx96.14[/tex]
[tex]\beta =\frac{n.\sum XY-\sum X.\sum Y}{n.\sum X^{2}-(\sum X)^{2}}=\frac{(9\times3732-(52\times722)}{(9\times460)-(52)^{2}} =-2.755\approx-2.76[/tex]
The regression equation is:
Final Grade = 96.14 - 2.76 Number of absence
For the value of Number of absence = 14 compute the value of Final grade as follows:
[tex]Final\ Grade = 96.14 - 2.76\ Number\ of\ absence\\=96.14-(2.76\times14)\\=57.5\\\approx58[/tex]
Thus, a student who was absent for 14 days received a final grade of 58.
To find the predicted final grade for 14 absences, calculate the slope and y-intercept of the regression line for the given data set to form the equation y=mx+b. With x as 14, solve the equation.
Explanation:To answer this question, we first need to find the equation of the regression line using the given number of absences (x) and final grades (y). This is achieved by calculating the slope and y-intercept of the best fit line for the data set. The formula for the slope (m) is given by the expression [N(Σxy) - (Σx)(Σy)] / [N(Σx^2) - (Σx)^2] and the y-intercept (b) by (Σy - m(Σx)) / N. After calculating these values, you can form the equation y = mx + b. Using the equation, input the absent times (14) into the x-variable to predict the final grade.
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Scarborough Faire Herb Farm is a small company specializing in selling organic fresh herbs, teas and herbal crafts. Currently, basil is their top selling herb, with $45,000 in sales last year. Parsley is their second biggest seller with $27,000 in sales. Total sales last year were $170,000 and Scarborough Faire forecasts sales to increase by 10% this year. If basil sales remain the same as last year but total sales grow as percentage will basil sales be this year?
Answer:
24.06% of total sales
Step-by-step explanation:
Total sales, which were $170,000 originally, are expected to grow by 10%. The expected value of total sales this year is:
[tex]S=\$170,000*1.10 = \$187,000[/tex]
If Basil sales remain at the same value of $45,000, the percentage of sales corresponding to basil is:
[tex]B=\frac{\$45,000}{\$187,000}*100\%\\B=24.06\%[/tex]
Therefore, basil will correspond to 24.06% of total sales.
Basil sales will represent approximately 24.06% of Scarborough Faire Herb Farm's projected total sales this year, given that total sales are forecasted to increase by 10% and basil sales remain unchanged.
Explanation:The student's question concerns the percentage of total sales that basil sales will represent for Scarborough Faire Herb Farm in the current year, assuming a 10% increase in total sales from the previous year and unchanged basil sales. Firstly, we calculate the projected total sales for this year by increasing last year's total sales by 10%. The calculation is as follows:
Total sales last year: $170,000Forecasted increase: 10%Projected total sales this year: $170,000 + ($170,000 × 0.10) = $170,000 + $17,000 = $187,000Since the basil sales are to remain the same as last year ($45,000), we now calculate what percentage this represents of the projected total sales for this year:
Basil sales this year: $45,000 (unchanged)Percentage of basil sales out of total sales: ($45,000 / $187,000) × 100%Final percentage: 24.06% (rounded to two decimal places)Thus, basil sales will account for approximately 24.06% of the Scarborough Faire Herb Farm's total sales this year.
Three students work independently on a homework problem. The probability that the first student solves the problem is 0.95. The probability that the second student solves the problem is 0.85. The probability that the third student solves the problem is 0.80. What is the probability that all are able to solve the problem
The probability that all three students solve the problem is calculated by multiplying their individual success probabilities together. The total probability in this case is 64.6%.
Explanation:Your question pertains to probability, a topic in Mathematics. When three students independently attempt to solve a problem, and you have the probabilities of their success, the probability that all three will successfully solve the problem is determined by the product of their respective probabilities.
Therefore, the probability that all three students - the first with a probability of 0.95, the second with 0.85, and the third with 0.80 - will successfully solve the problem is calculated as follows:
0.95 * 0.85 * 0.80 = 0.646
Hence, there is a 64.6% probability that all three students will successfully solve the problem.
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A builder of houses needs to order some supplies that have a waiting time Y for delivery, with a continuous uniform distribution over the interval from 1 to 4 days. Because she can get by without them for 2 days, the cost of the delay is fixed at $400 for any waiting time up to 2 days. After 2 days, however, the cost of the delay is $400 plus $50 per day (prorated) for each additional day. That is, if the waiting time is 3.5 days, the cost of the delay is $400 $50(1.5)
Find the expected value of the builder’s cost due to waiting for supplies.
Answer:
The Expected cosy of the builder is $433.3
Step-by-step explanation:
$400 is the fixed cost due to delay.
Given Y ~ U(1,4).
Calculating the Variable Cost, V
V = $0 if Y≤ 2
V = 50(Y-2) if Y > 2
This can be summarised to
V = 50 max(0,Y)
Cost = 400 + 50 max(0, Y-2)
Expected Value is then calculated by;
Waiting day =2
Additional day = at least 1
Total = 3
E(max,{0, Y - 2}) = integral of Max {0, y - 2} * ⅓ Lower bound = 1; Upper bound = 4, (4,1)
Reducing the integration to lowest term
E(max,{0, Y - 2}) = integral of (y - 2) * ⅓ dy Lower bound = 2; Upper bound = 4 (4,2)
E(max,{0, Y - 2}) = integral of (y) * ⅓ dy Lower bound = 0; Upper bound = 2 (2,0)
Integrating, we have
y²/6 (2,0)
= (2²-0²)/6
= 4/6 = ⅔
Cost = 400 + 50 max(0, Y-2)
Cost = 400 + 50 * ⅔
Cost = 400 + 33.3
Cost = 433.3
Answer:
the expected value of the builder’s cost due to waiting for supplies is $433.3
Step-by-step explanation:
Due to the integration symbol and also for ease of understanding, i have attached the explanation as an attachment.
onsider a random number generator designed for equally likely outcomes. If numbers between 0 and 99 are chosen, determine which of the following is not correct. a. If 100 numbers are generated comma each integer between 0 and 99 must occur exactly once. b. For each random number generated comma each integer between 0 and 99 has probability 0.01 of being selected. c. If a very large number of random numbers are generated comma then each integer between 0 and 99 would occur close to 1 % of the time. d. The cumulative proportion of times that a 0 is generated tends to get closer to 0.01 as the number of random numbers generated gets larger and larger.
Answer:
The following option is not correct:
(a) If 100 numbers are generated comma each integer between 0 and 99 must occur exactly once.
Step-by-step explanation:
This is not correct, as there is a possibility of an integer being generated twice when 100 numbers are generated.
This can be explained with an example such as stated below:
3 numbers are to be generated.
The number generated can either be 1, 2, or 3.
The probability for all three numbers to be generated once when the generator is run 3 times is:
(1/3)*(1/3)*(1/3) * Number of ways to arrange the three numbers
Thus this probability will be:
Probability = (1/3)^3 * 3!
Probability = 0.222
Since the probability here is not equal to 1, the probability for the same thing happening at a larger scale will also not be 1.
Final answer:
In a random number generator for numbers between 0 and 99, each integer should occur exactly once when 100 numbers are generated.
Explanation:
The correct statement among the options is:
a. If 100 numbers are generated, each integer between 0 and 99 must occur exactly once. This statement holds true in a random number generator designed for equally likely outcomes.Let's break it down:
Option a: To ensure each number between 0 and 99 appears once in 100 numbers, the generator should evenly distribute the outcomes.Option b: The probability of selecting each integer should indeed be 0.01 in an equally likely random number generator.Option c: With a large number of random numbers, each integer between 0 and 99 would indeed occur close to 1% of the time due to the even distribution.Option d: The cumulative proportion of selecting 0 getting closer to 0.01 is a characteristic of equally likely outcomes over a large number of trials.Which product is positive?
Identify the correct statements for the given functions of set {a, b, c, d} to itself.
1.
f(a) = b, f(b) = a, f(c) = c, f(d) = d is a one-to-one function as each element is an image of exactly one element.
2.
f(a) = b, f(b) = a, f(c) = c, f(d) = d is not a one-to-one function as a is mapped to b and b is mapped to a.
3.
f(a) = b, f(b) = b, f(c) = d, f(d) = c is a one-to-one function as each element is mapped to only one element.
4.
f(a) = b, f(b) = b, f(c) = d, f(d) = c is not a one-to-one function as b is an image of more than one elements.
5.
f(a) = d, f(b) = b, f(c) = c, f(d) = d is a one-to-one function as each element is mapped to only one element.
6.
f(a) = d, f(b) = b, f(c) = c, f(d) = d is not a one-to-one function as d is an image of more than one element.
Answer:
The correct answers are
option(1), option (4),option (5)
Step-by-step explanation:
One to one: Every image has exactly one unique pre-image in domain.
(1)
f(a) = b, f(b)=a, f(c)=c, f(d)=d
b has a pre-image in domain i.e a
a has a pre-image in domain i.e b
c has a pre-image in domain i.e c
d has a pre-image in domain i.e d
Here all elements have a unique pre- image in domain.
Therefore it is one to one.
(2)
f(a) = b, f(b)=a, f(c)=c, f(d)=d
b has a pre-image in domain i.e a
a has a pre-image in domain i.e b
c has a pre-image in domain i.e c
d has a pre-image in domain i.e d
Here all elements have a unique pre- image in domain.
Therefore it is one to one.
(3)
f(a) = b, f(b)=b, f(c)=d, f(d)=c
b has a pre-image in domain i.e a
b has a pre-image in domain i.e b
d has a pre-image in domain i.e c
c has a pre-image in domain i.e d
b has two pre image.
Here all elements have not a unique pre- image in domain.
Therefore it is not a one to one mapping.
(4)
f(a) = b, f(b)=b, f(c)=d, f(d)=c
b has a pre-image in domain i.e a
b has a pre-image in domain i.e b
d has a pre-image in domain i.e c
c has a pre-image in domain i.e d
b has two pre image.
Here all elements have not a unique pre- image in domain.
Therefore it is not a one to one mapping.
(5)
f(a) = d, f(b)=b, f(c)=d, f(d)=c
d has a pre-image in domain i.e a
b has a pre-image in domain i.e b
d has a pre-image in domain i.e c
c has a pre-image in domain i.e d
Here all elements have a unique pre- image in domain.
Therefore it is a one to one mapping.
(6)
f(a)=d, f(b)=b,f(c)=c,f(d)=d
d has a pre-image in domain i.e a
b has a pre-image in domain i.e b
c has a pre-image in domain i.e c
d has a pre-image in domain i.e d
Here all elements have a unique pre- image in domain.
Therefore it is a one to one mapping.
f(a) = b, f(b) = a, f(c) = c, f(d) = d and f(a) = d, f(b) = b, f(c) = c, f(d) = d are one-to-one function and f(a) = b, f(b) = b, f(c) = d, f(d) = c is not one-to-one function. Then the correct statements are 1, 4, and 5.
What is a function?The function is an expression, rule, or law that defines the relationship between one variable to another variable. Functions are ubiquitous in mathematics and are essential for formulating physical relationships.
One-to-one - every image has exactly one unique pre-image in the domain.
1. f(a) = b, f(b) = a, f(c) = c, f(d) = d
b has a pre image in a domian that is a
a has a pre image in a domian that is b
c has a pre image in a domian that is c
d has a pre image in a domian that is d
Here all elements have a unique pre-image in a domain.
Therefore it is one-to-one.
2. f(a) = b, f(b) = a, f(c) = c, f(d) = d
b has a pre image in a domian that is a
a has a pre image in a domian that is b
c has a pre image in a domian that is c
d has a pre image in a domian that is d
Here all elements have a unique pre-image in a domain.
Therefore it is one-to-one.
3. f(a) = b, f(b) = b, f(c) = d, f(d) = c
b has a pre image in a domian that is a
b has a pre image in a domian that is b
d has a pre image in a domian that is c
c has a pre image in a domian that is d
Here all elements do have not a unique pre-image in a domain.
Therefore it is not one-to-one.
4. f(a) = b, f(b) = b, f(c) = d, f(d) = c
b has a pre image in a domian that is a
b has a pre image in a domian that is b
d has a pre image in a domian that is c
c has a pre image in a domian that is d
Here all elements do have not a unique pre-image in a domain.
Therefore it is not one-to-one.
5. f(a) = d, f(b) = b, f(c) = c, f(d) = d
d has a pre image in a domian that is a
b has a pre image in a domian that is b
c has a pre image in a domian that is c
d has a pre image in a domian that is d
Here all elements have a unique pre-image in a domain.
Therefore it is one-to-one.
6. f(a) = d, f(b) = b, f(c) = c, f(d) = d
d has a pre image in a domian that is a
b has a pre image in a domian that is b
c has a pre image in a domian that is c
d has a pre image in a domian that is d
Here all elements have a unique pre-image in a domain.
Therefore it is one-to-one.
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Most analysts focus on the cost of tuition as the way to measure the cost of a college education. But incidentals, such as textbook costs, are rarely considered. A researcher at Drummand University wishes to estimate the textbook costs of first-year students at Drummand. To do so, she monitored the textbook cost of 250 first-year students and found that their average textbook cost was $300 per semester. Identify the population of interest to the researcher.
Answer:
The population of interest to the researcher were the 250 first-year students that were monitored.
Step-by-step explanation:
In descriptive statistics, the portion of the cost of college education to be determined and has been selected for analysis is calle d "sample", the sample the researcher is interested in, considers the textbooks cost of first-year students, therefore the 250 first-year students is the researcher´s population of interest. This method involved the collection, presentation, and characterization.
Linear differential equations sometimes occur in which one or both of the functions p(t) and g(t) for y′+p(t)y=g(t) have jump discontinuities. If t0 is such a point of discontinuity, then it is necessary to solve the equation separately for t < t0 and t > t0. Afterward, the two solutions are matched so that y is continuous at t0; this is accomplished by a proper choice of the arbitrary constants. The following problem illustrates this situation. Note that it is impossible also to make y′ continuous at t0.
Solve the initial value problem.
y' + 6y = g(t), y(0) = 0
where
g(t) = 1, 0 ≤ t ≤ 1,
= 0, t > 0.
For [tex]0\le t\le1[/tex], we have
[tex]y'+6y=1\implies e^{6t}y'+6e^{6t}=(e^{6t}y)'=e^{6t}\implies y=\dfrac16+Ce^{-6t}[/tex]
Given that [tex]y(0)=0[/tex], we have
[tex]0=\dfrac16+C\implies C=-\dfrac16[/tex]
so that
[tex]y=\dfrac16(1-e^{-6t})[/tex]
For [tex]t>1[/tex] (I think you mistakenly wrote [tex]t>0[/tex], which overlaps with the first definition of [tex]g(t)[/tex]), we have
[tex]y'+6y=0\implies e^{6t}y'+6e^{6t}y=(e^{6t}y)'=0\implies y=Ke^{-6t}[/tex]
We want this to be a continuation of the previously found solution [tex]y[/tex] at [tex]t=1[/tex], which means we need to pick [tex]K[/tex] such that
[tex]\dfrac16(1-e^{-6})=Ke^{-6}\implies K=\dfrac16(e^6-1)[/tex]
Then the solution to the IVP is
[tex]y(t)=\begin{cases}\frac16(1-e^{-6t})&\text{for }0\le t\le1\\\frac{e^6-1}6e^{-6t}&\text{for }t>1\end{cases}[/tex]
Alternatively, we can get around treating [tex]g(t)[/tex] piecemeal and resorting to the Laplace transform. Write
[tex]g(t)=\begin{cases}1&\text{for }0\le t\le1\\0&\text{for }t>1\end{cases}=u(t)-u(t-1)[/tex]
where
[tex]u(t-c)=\begin{cases}0&\text{for }t<c\\1&\text{for }t\ge c\end{cases}[/tex]
is the unit step function.
Take the Laplace transform of both sides of the ODE:
[tex]y'+6y=g(t)\overset{\text{L.T.}}{\implies}(sY-y(0))+6Y=\mathcal L_s\{g(t)\}[/tex]
where [tex]Y=Y(s)[/tex] is the Laplace transform of [tex]y(t)[/tex].
We have
[tex]\mathcal L_s\{g(t)\}=\displaystyle\int_0^\infty g(t)e^{-st}\,\mathrm dt=\int_0^1e^{-st}\,\mathrm dt=\dfrac{1-e^{-s}}s[/tex]
so that
[tex](s+6)Y=\frac{1-e^{-s}}s\implies Y=\dfrac{1-e^{-s}}{s(s+6)}=\dfrac{1-e^{-s}}6\left(\dfrac1s-\dfrac1{s+6}\right)[/tex]
Taking the inverse transform yields
[tex]y=\dfrac{1-u(t-1)}6-\dfrac{e^{-6t}}6(e^tu(t-1)-1)[/tex]
[tex]y=\dfrac{1-e^{-6t}}6+\dfrac{e^{6-6t}-1}6u(t-1)[/tex]
which is equivalent to the same solution found earlier; for [tex]0\le t\le1[/tex], [tex]u(t-1)=0[/tex], so that [tex]y=\frac{1-e^{-6t}}6[/tex]; for [tex]t>1[/tex], [tex]u(t-1)=1[/tex], and [tex]y=\frac{1-e^{-6t}}6+\frac{e^{6-6t}-1}6=\frac{(e^6-1)e^{-6t}}6[/tex].
The given differential equation needs to be solved separately for two time ranges because of the piecewise-defined function g(t). Solution for the corresponding equations are founded using the techniques of homogeneous equation solutions and the integrating factor method. These solutions are then matched at the point of continuity.
Explanation:The given differential equation is a first-order linear differential equation of the form y′+p(t)y=g(t). We need to solve this equation considering two cases due to the piecewise definition of g(t).
Case 1: For 0 ≤ t ≤ 1, g(t) = 1. The corresponding homogeneous equation is y' + 6y = 0, with the solution being y(t) = Ce-6t. We find the particular solution using the integrating factor method, yielding y(t) = t/6 - 1/36 + Ce-6t. Substituting the initial condition y(0) = 0 helps us solve for C, giving the final solution for this range as y(t) = t/6 - 1/36.
Case 2: For t > 1, g(t) = 0. The homogeneous solution is the same as in Case 1, but in this case, no particular solution needs to be added, so the solution is y(t) = Ce-6t. The constant is determined by making the function continuous at t=1. We ultimately get y(t) = (1-e-6(t-1))/36.
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Suppose you want to have $700,000 for retirement in 35 years. Your account earns 9% interest. How much would you need to deposit in the account each month?
Answer: you should deposit $236.2 each month.
Step-by-step explanation:
We would apply the formula for determining future value involving deposits at constant intervals. It is expressed as
S = R[{(1 + r)^n - 1)}/r][1 + r]
Where
S represents the future value of the investment.
R represents the regular payments made(could be weekly, monthly)
r = represents interest rate/number of payment intervals.
n represents the total number of payments made.
From the information given,
there are 12months in a year, therefore
r = 0.09/12 = 0.0075
n = 12 × 35 = 420
S = $700000
Therefore,
700000 = R[{(1 + 0.0075)^420 - 1)}/0.0075][1 + 0.0075]
700000 = R[{(1.0075)^420 - 1)}/0.0075][1.0075]
700000 = R[{(23.06 - 1)}/0.0075][1.0075]
700000 = R[{22.06}/0.0075][1.0075]
700000 = R[2941.3][1.0075]
700000 = 2963.36R
R = 700000/2963.36
R = 236.2
When the velocity v of an object is very large, the magnitude of the force due to air resistance is proportional to v squared with the force acting in opposition to the motion of the object. A shell of mass 2 kg is shot upward from the ground with an initial velocity of 600 m/sec. If the magnitude of the force due to air resistance is (0.1)v squared, when will the shell reach its maximum height above the ground? What is the maximum height? Assume the acceleration due to gravity to be 9.81 m divided by s squared.
The maximum height reached by the shell is approximately 18255.79 meters.
To find when the shell reaches its maximum height and the value of the maximum height, we need to consider the forces acting on the shell and analyze its motion.
1. Force due to gravity:
The force due to gravity is given by the formula:
Force_gravity = mass × acceleration_due_to_gravity
Here, the mass of the shell is 2 kg, and the acceleration due to gravity is 9.81 m/s².
2. Force due to air resistance:
The force due to air resistance is given by the formula:
Force_air_resistance = (0.1) × velocity²
Here, the velocity of the shell is given as 600 m/s.
Using Newton's second law, we can calculate the net force acting on the shell:
Net force = Force_gravity - Force_air_resistance
When the shell reaches its maximum height, the net force is equal to zero because there is no acceleration at that point. Therefore, we can set the net force equation to zero and solve for the time:
0 = Force_gravity - Force_air_resistance
mass × acceleration_due_to_gravity = (0.1) × velocity²
2 kg × 9.81 m/s² = (0.1) × (600 m/s)²
Simplifying further, we find:
19.62 = 0.1 × 360,000
Time = 600 m/s / 9.81 m/s²
Time ≈ 61.15 seconds
Therefore, the shell will reach its maximum height approximately 61.15 seconds after being shot upward.
To find the maximum height, we can use the kinematic equation:
h = v₀t - (1/2)gt²
Substituting the given values into the equation, we find:
h = (600 m/s) × (61.15 s) - (1/2) × (9.81 m/s²) × (61.15 s)²
h ≈ 18255.79 meters
Therefore, the maximum height reached by the shell is approximately 18255.79 meters.
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The shell will reach its maximum height after approximately 4.51 seconds and this maximum height is around 100 meters.
Explanation:This problem involves the physics of motion under the influence of gravity and air resistance. An important concept here is the 'terminal velocity', which is the velocity at which the upward force (due to the initial impulse provided to the shell) balances out the downward force (due to gravity and air resistance).
First, we must establish that the terminal velocity 'v' of the shell upwards will be achieved when the sum of the forces acting on it are zero. This gives us:
0 = -0.1v^2 + 2 * 9.81
. Solving this quadratic equation reveals v = sqrt(2*9.81/0.1) m/s ≈ 44.3 m/s.
After achieving this terminal velocity, the shell will start decelerating at a rate of 9.81 m/s^2 (the gravity acceleration). The time 't' that takes for the shell to stop moving upwards (so to reach its maximum height) can be calculated using the formula:
t = v / gravity = 44.3/9.81 ≈ 4.51 seconds
.
As for the maximum height 'h', it can be calculated using this formula:
h = v * t + 0.5 * (-gravity) * t^2
By inserting the values of v, gravity, and t in this equation, we find h ≈ 100 m.
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In a study of pain relievers, 50 people were given product A, and all but 11 experienced relief. In the same study, 100 people were given product B, and all but 14 experienced relief. Fill in the blanks of the statement below to make the statement the most reasonable possible. Produ. V ? performed worse in the study because % failed to get relief with this product, whereas only 6 failed to get relief with Product ?
Product A performed worse in the study because 22% failed to get relief with it, whereas only 14% failed to get relief with Product B.
In the given study of pain relievers, we need to determine which product performed worse based on the percentage of people who did not experience relief. For Product A, 50 people were given the product and all but 11 experienced relief. This means that 11 out of 50 people did not experience relief, so we calculate the failure rate as follows: (11/50) * 100 = 22%. For Product B, 100 people were given the product and all but 14 experienced relief, therefore the failure rate is: (14/100) * 100 = 14%.
With these failure rates, we can now fill in the blanks of the statement:
Product A performed worse in the study because 22% failed to get relief with this product, whereas only 14% failed to get relief with Product B.
A recent highway safety study found that in 65% of all accidents a driver was wearing a seatbelt. Accident reports indicated that 83% of those drivers escaped serious injury (defined as hospitalization or death), but only 49% of the non-belted drivers were so fortunate. Find the probability that a randomly selected driver was wearing a seatbelt, if this driver was not seriously injured. Show your work (if using notations, make sure to identify them). (Round your answer to 2 places after the decimal point).
Answer:
The probability that a randomly selected driver was wearing a seatbelt, if this driver was not seriously injured, that is, P(B|E) = 0.76
Step-by-step explanation:
Probability of wearing a seatbelt in an accident = P(B) = 65% = 0.65
Probability of not wearing a seatbelt in an accident = P(B') = 1 - 0.65 = 0.35
Probability of escaping hospitalization and/or death given that one is wearing a seatbelt = P(E|B) = 83% = 0.83
Probability of escaping hospitalization and/or death given that one isn't wearing a seatbelt = P(E|B') = 0.49
Find the probability that a randomly selected driver was wearing a seatbelt, if this driver was not seriously injured, that is, P(B|E)
The probability of P(X|Y) is given mathematically as P(X n Y)/P(Y)
P(B|E) = P(B n E)/P(E)
But P(E) is unknown at the moment.
But P(E) = P(B n E) + P(B' n E) mathematically,.
P(B n E) can be obtained using P(E|B) and P(B)
P(E|B) = P(B n E)/P(B)
P(B n E) = P(E|B) × P(B) = 0.83 × 0.65 = 0.5395
And
P(B' n E) can be obtained using P(E|B') and P(B')
P(E|B') = P(B' n E)/P(B')
P(B' n E) = P(E|B') × P(B') = 0.49 × 0.35 = 0.1715
P(E) = P(B n E) + P(B' n E) = 0.5395 + 0.1715 = 0.711
The probability that a randomly selected driver was wearing a seatbelt, if this driver was not seriously injured, that is, P(B|E)
P(B|E) = P(B n E)/P(E) = 0.5395/0.711 = 0.76
Terri and Donna both sell crafts at two different craft shows each weekend. Terri is charged a 5% commission on the amount of money she earns and pays $35 for her booth. Donna is charged a 3% commission on the amount of money she earns and pays $55 for her booth. On the last weekend in November, Terri and Donna both earned the same amount of money at their craft shows. They both paid their respective craft shows the same total amount of money for their booths and commission.
Set up a system of equations to model the amount of money Terri and Donna pay each weekend at the craft shows. Let x represent the money earned from sales, let T represent the total amount Terri pays in one weekend, and let D represent the total amount Donna pays in one weekend.
What is the solution to the system of equations found in Part A? Give your answer as an ordered pair.
What does the solution of the system of equations found in Part B represent in the context of this situation? Be sure to explain the meaning of the values in the solution.
Answer:
(x, T) = (x, D) = (1000, 85)each booth pays $85 in fees on rental and sales of $1000Step-by-step explanation:
A. Given
T = 0.05x +35 . . . . Terri's cost of operating a craft booth
D = 0.03x +55 . . . . Donna's cost of operating a craft boot
T = D
where x is the dollar amount of sales.
__
B. Solution
Subtracting the equation for D from that of T, we get ...
T - D = 0
(0.05x +35) -(0.03x +55) = 0 = 0.02x -20
0 = x -1000 . . . . . divide by 0.02
x = 1000
T = D = 0.05(1000) +35 = 85
(x, T) = (x, D) = (1000, 85)
__
C. Meaning
According to the given definitions of the variables, each booth pays a total of $85 in fees for sales of $1000.
The system of equations is T = 0.05x + 35 and D = 0.03x + 55. The solution is (1000, 85), meaning Terri and Donna each earned $1000 from sales and paid $85 total in booth and commission fees.
Explanation:The system of equations to model the amount of money Terri and Donna pay each weekend at the craft shows can be written as follows: T = 0.05x + 35 and D = 0.03x + 55.
Since we know that Terri and Donna both paid the same total amount of money for their booths and commission, it means T = D. Or, we can equate the two equations: 0.05x + 35 = 0.03x + 55. Solving for x gives us x = 1000. Substituting x = 1000 into T = 0.05x + 35 equation, we get T (and D) = 85. So, the solution to the system of equations is (1000, 85).
In the context of this situation, the solution means that Terri and Donna both earned $1000 from sales, and each paid $85 total for their booth and commission fees.
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A student wanted to construct a 95% confidence interval for the mean age of students in her statistics class. She randomly selected nine students. Their average age was 19.1 years with a sample standard deviation of 1.5 years. What is the best point estimate for the population mean? A. 1.5 years B. 19.1 years C. 9 years D. 2.1 years
Answer:
Option B) 19.1 years
Step-by-step explanation:
We are given the following in the question:
Sample size, n = 9
Sample mean, [tex]\bar{x}[/tex] = 19.1 years
Alpha, α = 0.05
Population standard deviation, σ = 1.5 years
We have to approximate best point estimate for population mean.
The best point estimate for population mean is the sample mean.
Thus, we can write
[tex]\mu = \bar{x} = 19.1[/tex]
Thus, the correct answer is
Option B) 19.1 years
Suppose the probability that a company will be awarded a certain contract is .25, the probability that it will be awarded a second contract is .21 and the probability that it will get both contracts is .13. What is the probability that the company will win at least one of the two contracts?
Answer:
33% probability that the company will win at least one of the two contracts
Step-by-step explanation:
We solve this problem building the Venn's diagram of these probabilities.
I am going to say that:
A is the probability that a company is awarded the first contract.
B is the probability that a company is awarded the second contract.
We have that:
[tex]A = a + (A \cap B)[/tex]
In which a is the probability that a company is awarded the first contract but not the second and [tex]A \cap B[/tex] is the probability that a company is awarded both contract.
By the same logic, we have that:
[tex]B = b + (A \cap B)[/tex]
The probability that it will get both contracts is .13.
This means that [tex]A \cap B = 0.13[/tex]
The probability that it will be awarded a second contract is .21
This means that [tex]B = 0.21[/tex]
[tex]B = b + (A \cap B)[/tex]
[tex]0.21 = b + 0.13[/tex]
[tex]b = 0.08[/tex]
The probability that a company will be awarded a certain contract is .25
This means that [tex]A = 0.25[/tex]
[tex]A = a + (A \cap B)[/tex]
[tex]0.25 = a + 0.13[/tex]
[tex]a = 0.12[/tex]
What is the probability that the company will win at least one of the two contracts?
[tex]A \cup B = a + b + A \cap B = 0.12 + 0.08 + 0.13 = 0.33[/tex]
33% probability that the company will win at least one of the two contracts
The probability that the company will win at least one of the two contracts is [tex]\(\frac{17}{50}\) or 0.34.[/tex]
To find the probability that the company will win at least one of the two contracts, we can use the principle of inclusion-exclusion. The principle states that the probability of the union of two events (in this case, winning either contract) is the sum of the probabilities of each event occurring individually, minus the probability of both events occurring together.
Let [tex]\(P(A)\)[/tex] be the probability that the company will win the first contract, [tex]\(P(B)\)[/tex] be the probability that the company will win the second contract, and [tex]\(P(A \cap B)\)[/tex] be the probability that the company will win both contracts. We are given:
[tex]\(P(A) = 0.25\), \(P(B) = 0.21\), \(P(A \cap B) = 0.13\).[/tex]
The probability that the company will win at least one contract, [tex]\(P(A \cup B)\)[/tex], is given by:
[tex]\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\][/tex]
Substituting the given probabilities:
[tex]\[P(A \cup B) = 0.25 + 0.21 - 0.13\] \[P(A \cup B) = 0.46 - 0.13\] \[P(A \cup B) = 0.33\][/tex]
To express this probability as a fraction, we can write 0.33 as [tex]\(\frac{33}{100}\)[/tex], which simplifies to[tex]\(\frac{17}{50}\)[/tex] when reduced to its simplest form.
Therefore, the probability that the company will win at least one of the two contracts is [tex]\(\frac{17}{50}\)[/tex] or 0.34
A light bulb has a lifetime that is exponential with a mean of 200 days. When it burns out a janitor replaces it immediately. In addition there is a handyman who comes at times of a Poisson process at rate .01 and replaces the bulb as "preventive maintenance." (a) How often is the bulb replaced? (b) In the long run what fraction of the replacements are due to failure?
Answer:
(a) The number of bulbs often replaces is 66.67.
(b) The fraction of the replacements that are due to failure, in the long run, is [tex]\frac{1}{3}[/tex].
Step-by-step explanation:
Let X = lifetime of a bulb and Y = time after which the bulb is replaced.
It is provided that X follows Exponential distribution with mean lifetime of a bulb is, 200 days.
And the rate at which the bulb is replaced is, 0.01 also following an Exponential distribution.
(a)
A bulb is replaced only after it burns out or a handyman comes at times of a Poisson process and replaces it.
Then min (X, Y) follows an Exponential distribution with parameter [tex](\frac{1}{200}+0.01)[/tex].
The mean of an Exponential distribution with parameter θ is:
[tex]Mean=\frac{1}{\theta}[/tex]
Compute the mean of min (X, Y) as follows:
[tex]Mean =\frac{1}{(\frac{1}{200}+0.01)} =\frac{1}{0.015}= 66.67[/tex]
Thus, the number of bulbs often replaces is 66.67.
(b)
Compute the probability of the event (X < Y) as follows:
[tex]P(X<Y)=\frac{0.005}{0.015} =\frac{1}{3}[/tex]
Thus, the fraction of the replacements that are due to failure, in the long run, is [tex]\frac{1}{3}[/tex].
Answer:
(a) The number of bulbs often replaces is 66.67.
(b) The fraction of the replacements that are due to failure, in the long run, is .
Step-by-step explanation:
Assume lim f(x)-6 and lim g(x)-9. Compute the following limit and state the limit laws used to justify the computation.
x→3 x →3
Lim 3√f(x).g(x)+10
x →3
Using the limit laws of addition and multiplication, the limit of the given equation is solved by combining the calculated limits of the function and the constant, giving an answer of approximately 13.78.
Explanation:To solve this problem, we can use the limit laws of addition and multiplication. The limit laws state that the limit of the sum of two functions is equal to the sum of their individual limits, and the limit of the product of two functions is equal to the product of their individual limits.
So, based on these laws, we first separate the function into two parts: the limit of 3√(f(x).g(x)) as x approaches 3 and the limit of 10 as x approaches 3.
Given lim f(x)=>6 and lim g(x)=>9 when x approaches 3, we multiply these values together to get a product of 54. The cubed root of 54 is approximately 3.78.
The constant 10 has a limit of 10, as constants maintain their value irrespective of the limit.
Adding these values together, 3.78 + 10, gives us a limit of approximately 13.78.
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The following information relates to Franklin Freightways for its first year of operations (data in millions of dollars): Pretax accounting income:$200 Pretax accounting income included: Overweight fines (not deductible for tax purposes) 5 Depreciation expense 70 Depreciation in the tax return using MACRS: 110 The applicable tax rate is 40%. There are no other temporary or permanent differences. Franklin Freightways experienced ($ in millions) a: Multiple Choice Tax liability of $66. Tax liability of $36. Tax liability of $70.6. Tax benefit of $10 due to the NOL.
Answer:
Franklin Freightways experienced $66 million.
Step-by-step explanation:
= ( Pretax accounting income + Overweight fines - Temporary difference: Depreciation) * tax rate
($200 + 5 - $40) × 40%
=Tax liability of $66.
If mABC= 184°, what is m∠ABC?
88°
90°
84°
92°
mADC=360-mABC=360-184=176⁰
So, measure of angle ABC= mADC/2=176/2=88⁰
The required measure of angle ABC is 88° for the given figure. The correct answer is option A.
What is an arc?The arc is a portion of the circumference of a circle. The circumference of a circle is the distance or perimeter around a circle
The measure of the arc is given as follows:
mABC = 184°
According to the given figure, mADC is the part of the full circle which is complete arc that measure of angle is 360°.
mADC = 360° - mABC
mADC = 360° - 184°
mADC = 176°
As we know that the measure of angle ABC is equal to half of mADC.
The measure of angle ABC = mADC/2
The measure of angle ABC = 176/2
The measure of angle ABC = 88°
Therefore, the required measure of angle ABC is 88°.
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A cupcake stand has 40 chocolate, 30 coconut and 20 banana cupcakes. Alice chooses 20 cupcakes at random to create a box as a present for her friend.
What is the probability that she chose:
(a) Eight banana and 6 coconut cupcakes?
(b) At least 2 chocolate cupcakes?
(c) All cupcakes of the same kind?
Answer:
a) 0.00563
b) 1
c) 0
Step-by-step explanation:
Total = 40+30+20 =90
a) (20C8×30C6×40C6)/90C20
= 0.00563
b) 1 - (no chocolate + 1 chocolate)
1 - [(50C20) + (40C1×50C19)]/90C20
1 - 0.00002478
= 0.9999752187
c) [40C20+20C20+30C20]/90C20
= 0.0000000027045
This is about permutations and combinations.
a) Probability = 0.00563
b) Probability = 0.99997522
c) Probability = 0.0000000027045
We are told cupcakes at the stand are;Chocolate = 40
Coconut = 30
Banana = 20
Total number of chocolates = 40 + 30 + 20
Total number of chocolates = 90
a) Probability that she will choose 8 banana and 6 chocolate cakes if she chooses 20 cupcakes at random will be;
(20C₈ × 30C₆ × 40C₆)/90C₂₀
(125970 × 593775 × 3838380)/50980740277700939310
This gives us 0.00563
b) Probability of at least 2 chocolate cupcakes is;
1 - [P(no chocolate) + P(1 chocolate)]
P(no chocolate) = (50C₂₀)/90C₂₀
P(1 chocolate) = (40C₁ × 50C₁₉)/90C₂₀
Thus;
1 - [P(no chocolate) + P(1 chocolate)] = 1 - [(40C₁ × 50C₁₉) + 50C₂₀]/90C₂₀
This gives us; 0.99997522
c) Probability of getting all cupcakes of same kind is;
(40C₂₀ + 20C₂₀ + 30C₂₀)/90C₂₀
⇒ 0.0000000027045
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marco and paolo are working with expressions with rational exponents. marco claims that 642/3=512.which of them is correct? use the rational exponent to justify your answer
Answer:
poalo is correct
Step-by-step explanation
Answer:
Neither Marco nor Paolo is correct, because neither one used the rational exponent property correctly.
Step-by-step explanation:
I just did the test and it told me that I got it right.
Hope that helps! Please mark me brainliest.
A study was made of seat belt use among children who were involved in car crashes that caused them to be hospitalized. It was found that children not wearing any restraints had hospital stays with a mean of 7.37 days and a standard deviation of 1.25 days with an approximately normal distribution. (a) Find the probability that their hospital stay is from 5 to 6 days, rounded to five decimal places. .10756 (b) Find the probability that their hospital stay is greater than 6 days, rounded to five decimal places.
Answer:
a) [tex]P(5<X<6)=P(\frac{5-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{6-\mu}{\sigma})=P(\frac{5-7.37}{1.25}<Z<\frac{6-7.37}{1.26})=P(-1.90<z<-1.10)[/tex]
And we can find this probability with this difference:
[tex]P(-1.90<z<-1.10)=P(z<-1.10)-P(z<-1.90)[/tex]
And using the norma standard distribution or excel we got:
[tex]P(-1.90<z<-1.10)=P(z<-1.10)-P(z<-1.90)=0.136-0.029=0.107[/tex]
b) [tex]P(X>6) =P(Z> \frac{6-7.37}{1.25}) = P(Z>-1.096)[/tex]
And using the complement rule we got:
[tex]P(Z>-1.096) =1-P(Z<-1.096) = 1-0.137= 0.863[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Part a
Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(7.37,1.25)[/tex]
Where [tex]\mu=7.37[/tex] and [tex]\sigma=1.25[/tex]
We are interested on this probability
[tex]P(5<X<6)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(5<X<6)=P(\frac{5-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{6-\mu}{\sigma})=P(\frac{5-7.37}{1.25}<Z<\frac{6-7.37}{1.26})=P(-1.90<z<-1.10)[/tex]
And we can find this probability with this difference:
[tex]P(-1.90<z<-1.10)=P(z<-1.10)-P(z<-1.90)[/tex]
And using the norma standard distribution or excel we got:
[tex]P(-1.90<z<-1.10)=P(z<-1.10)-P(z<-1.90)=0.136-0.029=0.107[/tex]
Part b
For this case we want this probability:
[tex] P(X>6)[/tex]
And we can use the z score and we got:
[tex]P(X>6) =P(Z> \frac{6-7.37}{1.25}) = P(Z>-1.096)[/tex]
And using the complement rule we got:
[tex]P(Z>-1.096) =1-P(Z<-1.096) = 1-0.137= 0.863[/tex]
To find the probability that the hospital stay is from 5 to 6 days, we need to standardize the values using the z-score formula. The probability that their hospital stay is from 5 to 6 days is approximately 0.10756. The probability that their hospital stay is greater than 6 days is approximately 0.86301.
Explanation:To find the probability that the hospital stay is from 5 to 6 days, we first need to standardize the values using the z-score formula.
z = (x - µ) / σ
Let's calculate the z-scores for x = 5 and x = 6.
For x = 5:
z = (5 - 7.37) / 1.25 = -1.896
For x = 6:
z = (6 - 7.37) / 1.25 = -1.096
Next, we can use the standard normal distribution table or a calculator to find the probabilities associated with these z-scores:
P(z < -1.896) = 0.02999
P(z < -1.096) = 0.13699
To find the probability that the hospital stay is from 5 to 6 days, we subtract P(z < -1.096) from P(z < -1.896):
P(5 < x < 6) = P(z < -1.896) - P(z < -1.096) = 0.02999 - 0.13699 = 0.10756
Therefore, the probability that their hospital stay is from 5 to 6 days is approximately 0.10756, rounded to five decimal places.
To find the probability that their hospital stay is greater than 6 days, we can use the standard normal distribution table or a calculator to find the probability associated with the z-score for x = 6:
P(z > -1.096) = 1 - P(z < -1.096) = 1 - 0.13699 = 0.86301
Therefore, the probability that their hospital stay is greater than 6 days is approximately 0.86301, rounded to five decimal places.
According to Ohm’s Law, the voltage V , current I , and resistance R in a circuit are related by the equation V = I R , where the units are volts, amperes, and ohms. Assume that the voltage is constant with V = 20 V. Calculate the average rate of change of I with respect to R for the interval from R = 6 to R = 6.1 . (Use decimal notation. Give your answer to three decimal places.)
Answer:
The average rate of change as follows of I is -0.185.
Step-by-step explanation:
The relation between voltage V, current I, and resistance R in a circuit, according to the Ohm's law is:
[tex]V = IR[/tex]
It is provided that:
V = 20 V
The interval between which R varies is, R = 8 to R = 8.1.
Compute the value of I as follows:
[tex]V=IR\Rightarrow I=\frac{I}{R}\Rightarrow I=\frac{20}{R}[/tex]
Compute the average rate of change as follows of I as follows:
[tex]\frac{\delta I}{\delta R}=\frac{(I\times8.1)-(I\times8)}{8.1-8}[/tex]
[tex]=\frac{1}{0.1}[\frac{12}{8.1}-\frac{12}{8}][/tex]
[tex]=\frac{12}{0.1}[\frac{8-8.1}{64.8}][/tex]
[tex]=-0.1852[/tex]
Thus, the average rate of change as follows of I is -0.185.