Answer: Lucas had $7
Step-by-step explanation:
Let x represent the amount of money that Lucas had initially.
Let y represent the weight of each type of oat that he bought.
Lucas bought a certain weight of oats for his horse at a unit price of .20 per pound. The total cost left him with one extra dollar. This means that
x - 0.2y = 1 - - - - - - - - - - - -- - 1
He wanted to buy the same weight for enriched oats instead, but at .30 per pound he was short 2 dollars. This means that
x - 0.3y = - 2- - - - - - - - - - - -- - 1
Subtracting equation 1 from equation 2, it becomes
0.1y = 3
y = 3/0.1
y = 30
Substituting y = 30 into equation 1, it becomes
x - 0.2 × 30 = 1
x - 6 = 1
x = 1 + 6
x = 7
Final answer:
By setting up equations based on the cost per pound of regular and enriched oats and the money Lucas had and was short, we deduce that Lucas wanted to buy 30 pounds of oats and he had $7 initially.
Explanation:
To solve for the amount of money Lucas had, we need to set up two separate equations based on the information given. If x is the weight of the oats in pounds and y is the total amount of money Lucas has, then:
Buying regular oats: 0.20x + 1 = yBuying enriched oats: 0.30x = y - 2Subtracting the first equation from the second gives us:
0.10x = 3
So, Lucas wanted to buy 30 pounds of oats. Substituting x back into either of the original equations gives us y:
0.20(30) + 1 = y6 + 1 = yy = 7Therefore, Lucas had $7 initially before making any purchase.
In k-means clustering, k represents the a. number of clusters. b. mean of the cluster. c. number of observations in a cluster. d. number of variables.
Answer:
The correct answer to the question is;
a. number of clusters.
Step-by-step explanation:
Clustering is the process of looking for smaller similar groups of observation within a set of data.
K-means clustering is a vector quantization method used in data mining cluster analysis. The objective of k-means clustering is to a given number of observations into k number of clusters whereby an observation is grouped in a cluster having the closest mean value, hence being representative of tha particular cluster. This is in atempt to make observations in a particular group to be similar.
In k-means clustering, the number of clusters is specified as k.
In k-means clustering, k represents the number of clusters, indicated by answer choice (a). This technique involves partitioning the data into k compact and separate clusters, with the k initial cluster centers often chosen randomly.
Explanation:In k-means clustering, k represents the number of clusters into which the data is to be partitioned. This method involves assigning each data point to the nearest cluster, while keeping the clusters as small as possible. The initial positions of the k clusters are typically chosen at random, and then the mean position of all the points in each cluster is recomputed, and this becomes the new center for the cluster. This process is repeated until the cluster assignments no longer change significantly, meaning the clusters are as compact and as separate as possible. The mean refers to the mean of the data points within each cluster once the clusters have formed. The standard deviation is a measure of the variability of the original distribution of the data. Sample size, denoted as n, is the number of observations in the dataset. Therefore, the answer to the question about what k represents in k-means clustering is (a) the number of clusters.
Write the quadratic function in standard form.
y = -(x + 2)^2
Answer:
Step-by-step explanation:
-(x+2)^2 --> -(x+2)(x+2) --> -(x^2+4x+4) = -x^2-4x-4
Answer:
[tex]x^{2} -4x+4[/tex]
Step-by-step explanation:
[tex]y =-(x+2)^{2}[/tex]
The negative sign multiplies the positive in the bracket
[tex]y=(x-2)^{2}[/tex]
[tex](x-2) X (x-2)[/tex]
[tex]x^{2} -2x-2x+4[/tex]
That gives us
[tex]x^{2} -4x+4[/tex]
Researchers are interested in learning more about the age of young adults who watch the television show Parks and Recreation. By interviewing people at a shopping mall, they can identify people who watch this show.
The mean age of these young adults at the mall who watch Parks and Recreation is an example of which of the following?
a. Parameter
b. Statistic
c. Population
d. Sample
Answer:
Option B) Statistic
Step-by-step explanation:
Parameter and Statistic
A parameter is a quantitative value that describes a population.A population is a collection of all possible observation for an event.A statistic is a quantitative variable that describes a sample.A sample is a part of population and is always smaller than the population.For the given research:
Population:
All young adults who watch the television show Parks and Recreation
Sample:
All young adults at the mall who watch the television show Parks and Recreation
Thus, the he mean age of these young adults at the mall who watch Parks and Recreation is a statistic as it describes a sample of the whole population.
Thus, the correct answer is :
Option B) Statistic
A political polling organization wishes to select a smaller focus group from a group of 7 Republicans, 10 Democrats, and 2 Independents. In how many ways can the group be chosen if: (a) it will consist of 1 Republican and 4 Democrats?(b) it will consist of 2 Republicans, 2 Democrats, and 3 Independents?
Final answer:
To determine the number of ways a smaller focus group can be chosen, we use combinations to calculate the possible selections for each political affiliation separately and then multiply these numbers. For part (a), there are 1470 ways to select 1 Republican and 4 Democrats. For part (b), there are 945 ways to choose 2 Republicans, 2 Democrats, and 1 Independent.
Explanation:
To find out how many ways a smaller focus group can be chosen from a larger group, we use combinations, which is a part of probability and combinatorics in mathematics.
Part (a): 1 Republican and 4 Democrats
To select 1 Republican out of 7, we calculate this as 7 choose 1, denoted as 7C1.
To select 4 Democrats out of 10, we calculate 10 choose 4, denoted as 10C4.
The total number of ways to choose the group is the product of these two combinations:
7C1 × 10C4 = 7 × (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 7 × 210 = 1470 ways
Part (b): 2 Republicans, 2 Democrats, and 1 Independent
Choosing 2 Republicans from 7, we have 7C2.
For the Democrats, 10C2. Since there are only 2 Independents, they are both selected, so no need to choose:
7C2 × 10C2 × 2C2 = (7 × 6) / (2 × 1) × (10 × 9) / (2 × 1) × 1 = 21 × 45 × 1 = 945 ways
What is data correlation. not anything too in depth just a over all definition
Tony collected 16.2 pounds of pecans from the trees on his farm.He will give the same weight of pecans to each of 12 friends.How many pounds of pecans will each friend get.
Each friend will get 1.35 pounds of pecans.
To find this, simply divide 16.2 by 12 to find the weight of pecans everyone gets. Thus making 1.35 pounds of pecans the answer.
I hope this helps!
The side length of a square is (6x-1) inches Write a linear expression in simplest form to represent the perimeter of the square.Find the perimeter of x equals 3
Final answer:
The perimeter of a square with side length (6x-1) is expressed as P = 24x - 4. Substituting x with 3, the perimeter is calculated to be 68 inches.
Explanation:
The expression for the perimeter of a square is given by the formula P=4s, where 's' is the side length of the square. For a square with side length (6x-1) inches, the perimeter would be:
P = 4(6x-1)
This expression can be simplified to:
P = 24x - 4
When x equals 3, we substitute 3 in place of x:
P = 24(3) - 4
P = 72 - 4
P = 68 inches
Therefore, the perimeter of the square when x is 3 is 68 inches.
5. We want to compare two different groups of students, students taking Composition 1 in a tradition lecture format and students taking Composition 1 in a distance learning format. We know that the mean score on the research paper is 85 for both groups. What additional information would be provided by knowing the standard deviation?
Answer:
By knowing the standard deviation, one gets the idea of how the value is scattered or dispersed about the mean.
Step-by-step explanation:
Let us first define standard deviation.
As it is known that the standard deviation is a measure of dispersion which express the spread of observation in terms of the average of deviations of observations from some central values.
Measure of dispersion gives us an idea about homogeneity or heterogeneity of the distribution.
Standard deviation is supposed almost an ideal measure of dispersion except the general nature of extracting the square root.
Thus for the given question, if we want to compare the two different groups of students whose mean score is 85. Here the standard deviation for both the groups interprets an idea about how the individual score for each group scattered or varied about the mean score i.e. 85.
The growth of a local raccoon population approximates a geometric sequence where an is the number of raccoons in a given year and n is the year. after 6 years there are 45 raccoons and after 8 years there are 71 raccoons.
Answer:
GENERAL EXPLICIT SEQUENCE IS GIVEN [tex]a_n = (14.74)(r)^{n-1)}[/tex]
Step-by-step explanation:
Let n be the number of year the data is recorded in.
a: The number of raccoons taken initially.
r: The multiplying factor
[tex]a_n[/tex] : The number of raccoon in the nth year.
As given: [tex]a_6 = 45, a_8 = 71[/tex]
Now, as the given situation can be expressed as GEOMETRIC SERIES:
[tex]a_n = a r^{(n-1)}[/tex]
Applying the same to given terms, we get:
[tex]a_6 = a r^{(6-1)} = ar^5 = 45\\\implies ar^5 = 45[/tex]
[tex]a_8 = a r^{(8-1)} = ar^7 =71\\\implies ar^7 = 71[/tex]
Dividing both equations, we get:
[tex]\frac{ar^7}{ar^5} = \frac{71}{45} \\\implies r^2 = 1.58\\\implies r = 1.25[/tex]
So, the first term [tex]a = \frac{45}{(1.25)^5} = 14 .74 \approx 15[/tex]
So, the GENERAL EXPLICIT SEQUENCE IS GIVEN as: [tex]a_n = (14.74)(r)^{n-1)}[/tex]
A theater ticket costs $20. The function h(x) = 20x represents the cost of purchasing x theater tickets. a. How much does it cost to buy 7 theater tickets? b. How many theater tickets can you buy with 460?
Answer:
A.) it costs 140$ for 7 tickets B.) 460$ = 23 tickets
Step-by-step explanation:
If it is 20 dolors for 1 ticket if you buy 7 you do 7 x 20 = 140.
But if you have 460$ then you do the opisit you do 460/20=23
I WILL GIVE A CROWN JUST NEED HELP ASAP
Answer:
C option is correct 219.
Step-by-step explanation:
Total fans = 365
Fans who bought popcorn = 3/5
No. of Fans who bought popcorn = 365 x 3/5
= 219
Answer:
Step-by-step explanation:
You have an SRS of 23 observations from a large population. The distribution of sample values is roughly symmetric with no outliers. What critical value would you use to obtain a 95% confidence interval for the mean of the population?
Answer:
Therefore the critical value= 2.073
Step-by-step explanation:
The number of observation = 23.
For the mean of the population the confidence interval = 95%.
Here mean and stander deviation of the distribution is not given. So we use t distribution.
T distribution is called as student's t distribution.
Sample number =n =23.
Confidence level = c= 95% =0.95
The degree of freedom is sample size decreased by 1
df=n-1 = 23-1 =22.
The critical value [tex]t^*[/tex] can be found in the row df = 22 and column with 0.95 of the T distribution table .
[tex]t^*[/tex] = 2.073
Therefore the critical value= 2.073
To calculate the critical value for a 95% confidence interval when the sample size is small (less than 30), we need to use the t-distribution. The critical value from the t-distribution will depend on the sample size (specifically, the degrees of freedom) and the desired level of confidence.
Here's how you could calculate this step by step without using the Python function mentioned:
**Step 1: Identify the desired confidence level.**
For a 95% confidence interval, we are interested in capturing the central 95% of the t-distribution.
**Step 2: Determine the degrees of freedom.**
The degrees of freedom (df) for a t-distribution is equal to the sample size minus 1. So with 23 observations, df = 23 - 1 = 22.
**Step 3: Find the critical t-value.**
We want to find the critical t-value for the t-distribution with 22 degrees of freedom that corresponds to the 95% confidence interval. This means we want to find the t-value such that 95% of the distribution lies between -t and +t. Because the t-distribution is symmetric, we can look up the critical value for 97.5% (to split the remaining 5% evenly on both tails of the distribution).
Using a t-distribution table (often found in the appendices of statistics textbooks) or a statistical computing resource, you would find the t-value that corresponds to a cumulative probability of 0.975 with 22 degrees of freedom.
**Step 4: Interpret the table or resource correctly.**
If you were looking at a table, you would look down the degrees of freedom column until you find 22, then right to the column that represents the 97.5% cumulative probability (remember, this is for the two-tailed test). That entry is the critical t-value that corresponds to a 95% confidence interval.
**Step 5: Use the critical t-value for constructing the interval.**
Once you have the critical t-value, you would use it to construct the confidence interval for the population mean by multiplying this t-value by the standard error of the sample mean and then add and subtract this value from the sample mean.
**Important Note:**
Please be aware that the exact t-value varies depending on the source of the statistical tables or the statistical software being used. The value also depends on the precision (number of decimals) presented in the table.
If you perform these steps with a standard statistical table or software, you should find that the critical t-value for a 95% confidence interval with 22 degrees of freedom is approximately 2.074.
A line segment that passes through the center and has endpoints on the circumference is called
Answer:
Diameter
Step-by-step explanation:
Knowing that a diameter is the largest chord as it passes through the center point of a circle, if the definition of the chord is a line segment with it's two endpoints on the circle (here by circle we mean circumference) then a diameter is a line segment that passes through the center and has endpoints on the circumference.
In a game, a player earns 100 points for each question answered correctly and earns −30 points for each question answered incorrectly. A player answered 14 questions correctly and 6 questions incorrectly. Write a numeric expression to represent the total number of points the player earned. What is the total number of points the player earned?
Answer:
1220
Step-by-step explanation:
Given that in a game, a player earns 100 points for each question answered correctly and earns −30 points for each question answered incorrectly.
Let x be the no of questions correctly answered . Then 20-x would be the question wrongly answered since total number of questions = 14+6 =20
Points gained for correct answer = 100(x) = 100x
Points lost for wrong answer = -30(20-x) = -600+30x
So total points gained when x questions are answered right
= [tex]100x-600+30x\\= 130x-600[/tex]
A player answered 14 questions correctly and 6 questions incorrectly.
Here x =14
Hence we substitute x =14 to get total points earned
Total points earned
[tex]= 130(14)-600\\= 1820-600\\=1220[/tex]
A satellite views the Earth at an angle of 20°. What is
the arc measure, x, that the satellite can see?
O 40°
O 80°
160°
0 320
Answer:
The answer is 160
Step-by-step explanation:
The value of x is (πr - 20).
What is the arc length of a circle?The arc length of a circle is the distance between two points on the curve of the circle.
We have,
The measure of an angle formed by two tangents outside the circle.
= Difference of the intercepted arc / 2 ______(1)
Now,
Larger arc = (2πr - x)
Small arc = x
Angle = 20
Substituting in (1).
20 = (2πr - x - x) / 2
40 = 2πr - 2x
20 = πr - x
x = πr - 20
Thus,
The value of x is (πr - 20).
Learn more about arc lengths here:
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Factor the expression. d2 – 4d + 4
(d + 2)2
(d – 4)(d – 1)
(d – 2)2
(d – 2)(d + 2)
The expression d^2 - 4d + 4 factors to (d - 2)^2.
To factor the expression d^2 + 4d + 4, we are looking for two binomials that will multiply together to give us the original quadratic expression. These binomials will be of the form (d - a)^2 because the last term is a perfect square (4 = 22) and the middle term is twice the product of the square roots of the first and last terms.
Here's the step-by-step factoring:
1. Identify the square root of the first term, which is d.
2. Identify the square root of the last term, which is 2.
3. Since our middle term is negative, we use negative signs in our binomials.
4. The factored form is (d - 2)^2, as this will expand to d^2 - 2*d*2 + 22, which simplifies to d^2 - 4d + 4.
a swimming pool is shaped like a cylinder with a radius of 15 feet and a height of 6 feet. if one cubic foot holds 7.48 gallons of the water how much gallons of water can the swimming pool hold.
Answer: the swimming pool can hold 31707.72 gallons of water.
Step-by-step explanation:
The swimming pool is shaped like a cylinder. The formula for determining the volume of a cylinder is expressed as
Volume = πr²h
Where
r represents the radius of the cylindrical swimming pool.
h represents the height of the pool.
π is a constant whose value is 3.14
Therefore, volume of the swimming pool is
Volume = 3.14 × 15² × 6 = 4239 cubic feet
if one cubic foot holds 7.48 gallons of the water, then the number of gallons of water that the swimming pool can hold is
4239 × 7.48 = 31707.72 gallons of water
Answer:
the swimming pool can hold 31707.72 gallons of water.
Step-by-step explanation:
At December 31, bonds payable of $109,993,000 are outstanding. The bonds pay 12% interest every September 30 and mature in installments of $27,498,250 every September 30, beginning September 30, 2018.
Answer:
Explanation:
If 112% interest of bond amount = $27,498,250
∴ 100% principal amount = 27,498,250 X 100/112 = $24,552,008.93
10% bond interest = $24,552,008.93 X 0.10 = $2,455,200.89
Between October 2018 and August 2019, it amounts = $2,455,200.09 X 11 = $27,007,209.82
The amount accrued up to September 2019 = $(27,007,209.82 + 27,498,250) = $54,505,459.82
From October 2019 to August 2020, it will amount = $(27,007,209.82 + 54,505,459.82) = $81,512669.64
The amount accrued up to September 2020 = $(81,512,669.64 + 27,498,250) = $109,010,919.64
Rachel gets her midterm grades and finds that she has a 2.4 in OB. She expected a better grade point average to date. Rachel is _________ her performance.
Answer:
Answer is; Evaluating
Step-by-step explanation:
Student self-assessment involves students evaluating their own work and learning progress.
Through self-assessment evaluation, students can:
* See where their knowledge is weak
* See where to focus their attention in learning.
* Set realistic goals
* Revise their work
* Track their own progress.
In Rachel's case, after writing her midterm exams, she already set her goal grade. So when she got her midterm grades, she needed to evaluate her work to find out her performance.
Therefore, according to the question, Rachel is EVALUATING her performance.
Given f(x) and g(x) = f(x + k), use the graph to determine the value of k.
Two lines labeled f of x and g of x. Line f of x passes through points negative 4, 0 and negative 2, 2. Line g of x passes through points negative 10, 0 and negative 8, 2.
6
−3
−6
3
Answer:
6
Step-by-step explanation:
f(-4) = g(-10) = f(-10+k)
f(-2) = g(-8) = f(-8+k)
-4 = -10+k
k = -4+10
k = 6
If the line f passes through (-4,0) and (-2,2) it means:
f(-4)=0 and f(-2)=2
If the line g passes through (-10,0) and (-8,2) it means:
g(-10)=0 and g(-8)=2
We can see that
g(-10)=0=f(-4)
g(-10)=0=f(-10+6)
Also,
g(-8)=2=f(-2)
g(-8)=2=f(-8+6)
Therefore, k=6
PLEASE HELP IM BEING TIMED
Answer:
The 3rd option Summation(4^i-4)
Step-by-step explanation:
Summation(4^i-4)
When i = 1
4^I-4 = 4^1-4 = 4^-3 = 1/64
When i = 2
4^i-4 = 4^2-4 = 4^-2 = 1/16
When i = 3
4^i-4 = 4^3-4 = 4^-1 = 1/4
When i = 4
4^i-4 = 4^4-4 = 4^0 = 1
When i = 5
4^i-4 = 4^5-4 = 4^1 = 4
Answer:
[tex]\sum _{i=1}^54^{i-4}[/tex]
Step-by-step explanation:
[tex]a_{1} \\[/tex] = First term
In this case, our first term is [tex]\frac{1}{64}[/tex]
The ratio of all of the adjacent terms is 4
[tex]a_{1} \\[/tex] = [tex]4^1^-^4[/tex]
[tex]a_{1}=\frac{1}{64}[/tex]
~Hope this helps!~
Find the perimeter of the polygon if B = D
Perimeter of the polygon is 92 cm
Solution:
The reference image for the solution is attached below.
AX = 10.5 cm, BY = 11.5 cm and CZ = 12.5 cm
AW and AX are tangents to the circle from external point A.
BX and BY are tangents to the circle from external point B.
CY and CZ are tangents to the circle from external point C.
DZ and DW are tangents to the circle from external point D.
Two tangents drawn from an external point to a circle are equal in length.
AW = AX, BX = BY, CY = CZ and DZ = DW
⇒ AW = AX
AW = 10.5 cm
⇒ BX = BY
BX = 11.5 cm
⇒ CY = CZ
CY = 12.5 cm
Given ∠B ≅ ∠D
If two angles are congruent, then the corresponding sides are congruent.
BX = BY = DZ = DW
DZ = DW = 11.5 CM
Perimeter = AW + AX + BX + BY + CY + CZ + DZ + DW
= 10.5 + 10.5 + 11.5 + 11.5 + 12.5 + 12.5 + 11.5 + 11.5
= 92
Perimeter of the polygon is 92 cm.
dont skip help me plzzz will mark brainliest
Answer:
(6,-6)
Step-by-step explanation:
well if you count over to the right 6 and down 6 that points would be
(6,-6)
Answer:
(6,-6)
Step-by-step explanation:
The answer is that because the x-axis for A is positive and it is 6 points away from the origin. The y-axis for A should negative this is because the quadrants are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the (x; y) coordinates are I (+; +), II (−; +), III (−; −), and IV (+; −). The point is also 6 units away from the origin.
What would be the results after the following code was executed? int[] x = {23, 55, 83, 19}; int[] y = {36, 78, 12, 24}; for(int a = 0; a < x.length; a++) { x[a] = y[a]; y[a] = x[a]; }
Answer:
x = y = {36, 78, 12, 24}
Step-by-step explanation:
The loop executes 4 times, as indicated by the length of array x.
The first line in the content of the loop assigns every element in array y to array x. Because both arrays now have the same content, the second line of code is quite redundant and is assigning the new values of x to y. Since these new values of x are the old values of y, there is no change in the contents of y. They are just being replaced by themselves.
In other words, the second line is not needed for anything. In fact, if the loop has much more contents, the second makes it work twice as much, reducing efficiency.
Jamie is riding a Ferris wheel that takes fifteen seconds for each complete revolution. The diameter of the wheel is 10 meters and its center is 6 meters above the ground. (a) When Jamie is 9 meters above the ground and rising, at what rate (in meters per second) is Jamie gaining altitude? (b) When is Jamie rising most rapidly? At what rate?
Answer:
The answers to the question is
(a) Jamie is gaining altitude at 1.676 m/s
(b) Jamie rising most rapidly at t = 15 s
At a rate of 2.094 m/s.
Step-by-step explanation:
(a) The time to make one complete revolution = period T = 15 seconds
Here will be required to develop the periodic motion equation thus
One complete revolution = 2π,
therefore the we have T = 2π/k = 15
Therefore k = 2π/15
The diameter = radius of the wheel = (diameter of wheel)/2 = 5
also we note that the center of the wheel is 6 m above ground
We write our equation in the form
y = [tex]5*sin(\frac{2*\pi*t}{15} )+6[/tex]
When Jamie is 9 meters above the ground and rising we have
9 = [tex]5*sin(\frac{2*\pi*t}{15} )+6[/tex] or 3/5 = [tex]sin(\frac{2*\pi*t}{15} )[/tex] = 0.6
which gives sin⁻¹(0.6) = 0.643 =[tex]\frac{2*\pi*t}{15}[/tex]
from where t = 1.536 s
Therefore Jamie is gaining altitude at
[tex]\frac{dy}{dt} = 5*\frac{\pi *2}{15} *cos(\frac{2\pi t}{15}) =[/tex] 1.676 m/s.
(b) Jamie is rising most rapidly when the velocity curve is at the highest point, that is where the slope is zero
Therefore we differentiate the equation for the velocity again to get
[tex]\frac{d^2y}{dx^2} = -5*(\frac{\pi *2}{15} )^2*sin(\frac{2\pi t}{15})[/tex] =0, π, 2π
Therefore [tex]-sin(\frac{2\pi t}{15} )[/tex] = 0 whereby t = 0 or
[tex]\frac{2\pi t}{15}[/tex] = π and t = 7.5 s, at 2·π t = 15 s
Plugging the value of t into the velocity equation we have
[tex]\frac{dy}{dt} = 5*\frac{\pi *2}{15} *cos(\frac{2\pi t}{15}) =[/tex] - 2/3π m/s which is decreasing
so we try at t = 15 s and we have [tex]\frac{dy}{dt} = 5*\frac{\pi *2}{15} *cos(\frac{2\pi *15}{15}) = \frac{2}{3} \pi[/tex]m/s
Hence Jamie is rising most rapidly at t = 15 s
The maximum rate of Jamie's rise is 2/3π m/s or 2.094 m/s.
(a) When Jamie is 9 meters above the ground and rising, she is gaining altitude at approximately 1.68 meters per second. (b) Jamie is rising most rapidly when the cosine function is at its maximum, which happens at the lowest point of the Ferris wheel, and the rate is approximately 2.094 meters per second.
Part (a): Rate at Which Jamie is Gaining Altitude
1. Identify the position function of Jamie on the Ferris wheel:
The height h of Jamie above the ground as a function of time t can be modeled by the equation of a sinusoidal function:
[tex]\[ h(t) = 6 + 5\sin\left(\frac{2\pi}{15}t\right) \][/tex]
Here, 6 meters is the height of the center of the Ferris wheel above the ground, and 5 meters is the radius of the wheel.
2. Differentiate the height function to find the rate of change of height:
To find the rate at which Jamie is gaining altitude, we need to differentiate h(t) with respect to t:
[tex]\[ h'(t) = \frac{d}{dt} \left( 6 + 5\sin\left(\frac{2\pi}{15}t\right) \right) = 5 \cdot \frac{2\pi}{15} \cos\left(\frac{2\pi}{15}t\right) \][/tex]
Simplifying,
[tex]\[ h'(t) = \frac{2\pi}{3} \cos\left(\frac{2\pi}{15}t\right) \][/tex]
3. Determine t when Jamie is at 9 meters above the ground and rising:
[tex]\[ 9 = 6 + 5\sin\left(\frac{2\pi}{15}t\right) \] Solving for \( \sin \left(\frac{2\pi}{15}t\right) \): \[ 3 = 5\sin\left(\frac{2\pi}{15}t\right) \] \[ \sin\left(\frac{2\pi}{15}t\right) = \frac{3}{5} \][/tex]
Jamie is rising when [tex]\( \cos \left(\frac{2\pi}{15}t\right) > 0 \)[/tex].
4. Find the rate at which Jamie is gaining altitude at this instant:
Substitute [tex]\(\sin \left(\frac{2\pi}{15}t\right) = \frac{3}{5}\)[/tex] into the derivative [tex]\( h'(t) \)[/tex]:
[tex]\[ \cos \left(\frac{2\pi}{15}t\right) = \sqrt{1 - \sin^2 \left(\frac{2\pi}{15}t\right)} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{\frac{16}{25}} = \frac{4}{5} \][/tex]
Thus, the rate of change of height:
[tex]\[ h'(t) = \frac{2\pi}{3} \cdot \frac{4}{5} = \frac{8\pi}{15} \approx 1.68 \text{ meters per second} \][/tex]
Part (b): When Jamie is Rising Most Rapidly
1. Identify when Jamie is rising most rapidly:
Jamie rises most rapidly when [tex]\( \cos \left(\frac{2\pi}{15}t\right) = 1 \)[/tex], which corresponds to the maximum value of the cosine function.
2. Rate of change of height at maximum rise:
[tex]\[ h'(t) = \frac{2\pi}{3} \cdot 1 = \frac{2\pi}{3} \approx 2.094 \text{ meters per second} \][/tex]
In 2000 the population of a country reached 1 billion, and in 2025 it is projected to be 1.2 billion. (a) Find values for C and a so that P(x)equalsCa Superscript x minus 2000 models the population of a country in year x. (b) Estimate the country's population in 2010. (c) Use P to determine the year when the country's population might reach 1.4 billion. (a) Cequals nothing (Type an integer or decimal rounded to five decimal places as needed.)
Answer:
(a) The value of C is 1.
(b) In 2010, the population would be 1.07555 billions.
(c) In 2047, the population would be 1.4 billions.
Step-by-step explanation:
(a) Here, the given function that shows the population(in billions) of the country in year x,
[tex]P(x)=Ca^{x-2000}[/tex]
So, the population in 2000,
[tex]P(2000)=Ca^{2000-2000}[/tex]
[tex]=Ca^{0}[/tex]
[tex]=C[/tex]
According to the question,
[tex]P(2000)=1[/tex]
[tex]\implies C=1[/tex]
(b) Similarly,
The population in 2025,
[tex]P(2025)=Ca^{2025-2000}[/tex]
[tex]=Ca^{25}[/tex]
[tex]=a^{25}[/tex] (∵ C = 1)
Again according to the question,
[tex]P(2025)=1.2[/tex]
[tex]a^{25}=1.2[/tex]
Taking ln both sides,
[tex]\ln a^{25}=\ln 1.2[/tex]
[tex]25\ln a = \ln 1.2[/tex]
[tex]\ln a = \frac{\ln 1.2}{25}\approx 0.00729[/tex]
[tex]a=e^{0.00729}=1.00731[/tex]
Thus, the function that shows the population in year x,
[tex]P(x)=(1.00731)^{x-2000}[/tex] ...... (1)
The population in 2010,
[tex]P(2010)=(1.00731)^{2010-2000}=(1.00731)^{10}=1.07555[/tex]
Hence, the population in 2010 would be 1.07555 billions.
(c) If population P(x) = 1.4 billion,
Then, from equation (1),
[tex]1.4=(1.00731)^{x-2000}[/tex]
[tex]\ln 1.4=(x-2000)\ln 1.00731[/tex]
[tex]0.33647 = (x-2000)0.00728[/tex]
[tex]0.33647 = 0.00728x-14.56682[/tex]
[tex]0.33647 + 14.56682 = 0.00728x[/tex]
[tex]14.90329 = 0.00728x[/tex]
[tex]\implies x=\frac{14.90329}{0.00728}\approx 2047[/tex]
Therefore, the country's population might reach 1.4 billion in 2047.
Tarun has 4 more than the twice the number of tshirts Deepak has.Mahesh has 2 more than thrice the number of T-shirts that Tarun has.If the ratio is 6:7 find the number of T-shirts each of them has.
Answer:
Deepak: 4 t-shirts,
Tarun: 12 t-shirts,
Mahesh: 14 t-shirts.
Step-by-step explanation:
Let x represent number of t-shirts that Deepak has.
Please consider the complete question.
Tarun has 4 t-shirt more than twice the number of T-shirts Deepak has. Mahesh has 2 more than thrice the number of T-shirts Deepak has. If the ratio of the t-shirt that Tarun and Mahesh have is 6 : 7, find out the number of the t-shirt each of them has.
Since Tarun has 4 t-shirt more than twice the number of T-shirts Deepak has, so the number of t-shirts that Tarun has would be [tex]2x+4[/tex].
We are also told that Mahesh has 2 more than thrice the number of T-shirts Deepak has. So the number of t-shirts that Mahesh has would be [tex]3x+2[/tex].
Since the ratio of the t-shirt that Tarun and Mahesh have is 6 : 7, so we can represent this information in an equation as:
[tex]\frac{2x+4}{3x+2}=\frac{6}{7}[/tex]
Cross multiply:
[tex]6(3x+2)=7(2x+4)[/tex]
[tex]18x+12=14x+28[/tex]
[tex]18x-14x+12-12=14x-14x+28-12[/tex]
[tex]4x=16[/tex]
[tex]x=\frac{16}{4}=4[/tex]
Therefore, Deepak has 4 t-shirts.
The number of t-shirts that Tarun has would be [tex]2x+4\Rightarrow 2(4)+4=4+4=12[/tex]
Therefore, Tarun has 12 t-shirts.
The number of t-shirts that Mahesh has would be [tex]3x+2\Rightarrow 3(4)+2=12+2=14[/tex]
Therefore, Mahesh has 14 t-shirts.
Joyce knits baby sweaters and baby socks. The baby sweaters take 10 feet of yarn and the baby socks take 5 feet of yarn. She has 100 feet of yarn and wants to make 5 sweaters. What is the maximum number of socks she will be able to make from the leftover yarn?
Let x represent sweaters and y represent socks.
Select one:
A. 8
B. 9
C. 11
D. 10
Answer: the maximum number of socks she will be able to make from the leftover yarn is 10
Step-by-step explanation:
Let x represent the number of sweaters.
Let y represent the number of socks.
The baby sweaters take 10 feet of yarn and the baby socks take 5 feet of yarn. This means that the total number of feet of yarn needed to make x sweaters and y socks is expressed as
10x + 5y
She has 100 feet of yarn and wants to make 5 sweaters. This means that
10 × 5 + 5y = 100
50 + 5y = 100
5y = 100 - 50 = 50
y = 50/5
y = 10
Jocelyn estimates that a piece of wood measures 5.5 cm. If it actually measures 5.62 cm, what is the percent error of Jocelyn's estimate?2.13%2.18%12%46.83%
Answer:
is d
Step-by-step explanation:
Answer:
2.13%
Step-by-step explanation:
Inaccurate measurement = 5.5cm
Actual measurement = 5.62cm
Difference= actual - inaccurate
=> 5.62 - 5.5
Therefore, the measurement Jocelyn obtained is off by 0.12cm (difference)
Note: error = difference
% diff = error÷ actual measurement × 100
= 0.12/5.62 × 100
% diff. = 2.13 (to the nearest decimal place)
Consider the continuous random variable x, which has a uniform distribution over the interval from 20 to 28. The probability density function has what value in the interval between 20 and 28?a. 1.000 b. 0c. 0.125 d. 0.050
Answer:
The probability density function is f(x)=0.125.
Step-by-step explanation:
We have he continuous random variable x, which has a uniform distribution over the interval from 20 to 28. We calculate the probability density function has what value in the interval between 20 and 28.
We use the formula:
[tex]\boxed{f(x)=\frac{1}{b-a}}[/tex]
We have a=20 and b=28, we get
[tex]f(x)=\frac{1}{28-20}\\\\f(x)=\frac{1}{8}\\\\f(x)=0.125\\[/tex]
The probability density function is f(x)=0.125.
The value of the probability density function (pdf) in the interval between 20 and 28 is 0.125.
Explanation:The probability density function (pdf) of a continuous uniform distribution is represented by a horizontal line. Since the random variable x has a uniform distribution over the interval from 20 to 28, the pdf will also have a constant value over this interval.
To find the value of the pdf in the interval between 20 and 28, we need to calculate the area under the pdf curve in this interval. Since the pdf is a horizontal line, the area is simply equal to the height multiplied by the width of the interval.
Since the width of the interval is 8 (28 - 20) and the pdf is a uniform distribution, the height is 1/8. Therefore, the value of the pdf in the interval between 20 and 28 is 1/8 or 0.125.
Learn more about Probability density function here:https://brainly.com/question/35625239
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