Answer:
dQ/dt = 3 - 2Q/(t+200), Q(0) = 100
Step-by-step explanation:
The rate of change dQ/dt describes the amount of salt in the tank at a given time, this rate of change can be express as rate-in minus the rate-out and those individual rates can be expressed as the rate at which liquid enters or leaves the tank (gal/min) by the amount of salt entering or leaving the tank at a given time (lb/gal).
Rate-in: the amount of liquid entering the tank per minute remains constant at 3 gal/min, and the amount of salt per gallon remains constant as well at 1 lb/gal, therefore, the rate-in is r1 = 3 gal/min * 1 lb/gal = 3 lb/minRate-out: the amount of liquid leaving the tank per minute remains constant at 2 gal/min but the amount of salt per gallon does not because the concentration of the salt is constantly changing due to the entering and leaving liquid, to calculate the variable amount of salt per gallon we divide the amount of salt in the tank Q at a given time by the volume of liquid at given time, this volume can be calculated as the initial content of liquid (200 gal) plus the net amount of liquid entering the tank (liquid entering the tank minus liquid leaving it) multiplied by time, the volume at a given time is ((3 gal/min - 2 gal/min)*t + 200 gal), therefore, the rate-out is r2 = 2 gal/min * Q/(t*(1 gal/min)+200)
Collecting everything said:
dQ/dt = r1 - r2 = 3 lb/min - 2 gal/min * Q/(t*(1 gal/min)+200)
Note:
The units are really important when solving the equation As you can see from our result the amount of solution in the tank depends on the timeThe initial condition Q(0) = 100 describe the amount of salt at t = 0 and is fundamental to solve the differential equation
In five-card poker, a straight consists of five cards with adjacent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king of clubs). Assuming that aces can be high or low, if you are dealt a five-card hand, what is the probability that it will be a straight with high card 9? (Round your answer to six decimal places.)
Answer:
[tex]0.000394[/tex]
Step-by-step explanation:
First we will find the probability of selecting five cards out of pack of cards
Probability of selecting five cards is equal to
[tex]^{52}C_5[/tex]
On expanding we get
[tex]\frac{52!}{47! * 5!} \\[/tex]
[tex]\frac{52 * 51 * 50 * 49 * 48 * 47!}{47 ! * 5*4*3*2*1} \\= 2598960[/tex]
straight high card [tex]9[/tex] means five cards with values lesser than [tex]9[/tex] but adjacent to it are
[tex]9, 8, 7, 6, 5[/tex]
there are four card for each number
Hence, probability of choosing five cards is equal to
[tex]4*4*4*4*4\\= 1024[/tex]
Probability of getting a straight with high card 9 is equal to
[tex]\frac{1024}{2598960}[/tex]
[tex]0.000394[/tex]
what is the solution to the equation below? round your answer to two decimal places. 4 • 8^x=11.48
Answer:
x = 0.51
Step-by-step explanation:
[tex]4\cdot 8^x = 11.48\\8^x = \frac{11.48}{4}\\ 8^x = 2.87\\\log_8 8^x = \log_8 2.87\\x=\log_8 2.87\\x=0.51[/tex]
Answer:
x=0.51
Step-by-step explanation:
A painter is placing a ladder to reach the third story window, which is 19 feet above the ground and makes an angle with the ground of 80. How far out from the building does the base of the latter need to be positioned? Round your answer to the nearest 10th. The base of the latter needs to be positioned__ feet out from the building
Answer:
The answer to your question is 3.35 ft
Step-by-step explanation:
Data
height = 19 ft
angle = 80°
Process
1.- It is formed a right triangle so use a trigonometric function that relates the opposite side and the adjacent side. This trigonometric function is tangent.
tan Ф = Opposite side/adjacent side
adjacent side = Opposite side / tan Ф
adjacent side = 19 / tan 80
adjacent side = 19 / 5.67
adjacent side = 3.35 ft
Answer: The base of the latter needs to be positioned 3.6 feet out from the building.
Step-by-step explanation:
The ladder forms a right angle triangle with the building and the ground. The length of the ladder represents the hypotenuse of the right angle triangle. The height from the where the top of the ladder touches the window to the base of the building represents the opposite side of the right angle triangle.
The distance from the bottom of the ladder to the base of the building represents the adjacent side of the right angle triangle.
To determine distance,h from the bottom of the ladder to the base of the building, we would apply
the tangent trigonometric ratio.
Tan θ = opposite side/adjacent.
Tan 80 = 19/h
h = 19/Tan 80 = 19/5.6713
h = 3.6 feet
The Consumer Reports National Research Center conducted a telephone survey of 2000 adults to learn about the major economic concerns for the future (Consumer Reports, January 2009). The survey results showed that 1760 of the respondents think the future health of Social Security is a major economic concern.
a. What is the point estimate of the population proportion of adults who think the future health of Social Security is a major economic concern (to 2 decimals)?
b. At 90% confidence, what is the margin of error (to 4 decimals)?
c. Develop a 90% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern (to 3 decimals).(, )
d. Develop a 95% confidence interval for this population proportion (to 4 decimals).
Answer:
a) [tex] \hat p =\frac{X}{n}=\frac{1760}{2000}=0.88[/tex]
b) [tex]ME=1.64* \sqrt{\frac{0.88*(1-0.88)}{2000}}=0.0119[/tex]
c) [tex]0.88 - 1.64 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.868[/tex]
[tex]0.88 + 1.64 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.892[/tex]
And the 90% confidence interval would be given (0.868;0.892).
d) [tex]0.88 - 1.96 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.8658[/tex]
[tex]0.88 + 1.96 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.8942[/tex]
And the 90% confidence interval would be given (0.8658;0.8942).
Step-by-step explanation:
Part a
For this case the point of estimate for the population proportion is given by:
[tex] \hat p =\frac{X}{n}=\frac{1760}{2000}=0.88[/tex]
Part b
The confidence interval would be given by this formula
[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
For the 90% confidence interval the value of [tex]\alpha=1-0.9=0.1[/tex] and [tex]\alpha/2=0.05[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=1.64[/tex]
The margin of error is given by:
[tex] ME=z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
And if we replace we got:
[tex]ME=1.64* \sqrt{\frac{0.88*(1-0.88)}{2000}}=0.0119[/tex]
Part c
And replacing into the confidence interval formula we got:
[tex]0.88 - 1.64 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.868[/tex]
[tex]0.88 + 1.64 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.892[/tex]
And the 90% confidence interval would be given (0.868;0.892).
Part d
For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=1.96[/tex]
[tex]0.88 - 1.96 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.8658[/tex]
[tex]0.88 + 1.96 \sqrt{\frac{0.88(1-0.88)}{2000}}=0.8942[/tex]
And the 90% confidence interval would be given (0.8658;0.8942).
The Main Answer for:
a. The point estimate of the population proportion of adults who think the future health of Social Security is a major economic concern is approximately [tex]0.88[/tex].
b. The margin of error at [tex]90[/tex]% confidence is approximately [tex]0.0120[/tex].
c. The [tex]90[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern is approximately ([tex]0.868, 0.892[/tex]).
d. The [tex]95[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern is approximately ([tex]0.8657, 0.8943[/tex]).
a. Point Estimate: The point estimate of the population proportion can be calculated by dividing the number of respondents who think the future health of Social Security is a major economic concern by the total number of respondents surveyed.
Point Estimate = Number of respondents concerned about Social Security / Total number of respondents
Given: Number of respondents concerned about Social Security = [tex]1760[/tex]
Total number of respondents surveyed = [tex]2000[/tex]
Point Estimate = [tex]1760 / 2000 =0.88[/tex]
So, the point estimate of the population proportion of adults who think the future health of Social Security is a major economic concern is approximately [tex]0.88[/tex].
b. Margin of Error: The margin of error (E) can be calculated using the formula:
[tex]E=z*\sqrt{p(1-p)/n}[/tex]
where:
• z is the z-score corresponding to the desired confidence level
• p is the point estimate of the population proportion
• n is the sample size
Since we are aiming for a 90% confidence interval, we find the z-score corresponding to a 90% confidence level, which is approximately 1.645 (you can find this value in a standard normal distribution table).
[tex]E=1.645*\sqrt{0.88(1-0.88)/ 2000} \\E=1.645*\sqrt{0.88*0.12/2000} \\E=1.645*\sqrt{0.1056/2000} \\E=1.645*\sqrt{0.0000528} \\E=1.645*0.00727\\E=0.01196[/tex]
So, the margin of error at [tex]90[/tex]% confidence is approximately [tex]0.0120[/tex].
c. Confidence Interval (90%): The confidence interval can be calculated using the point estimate and the margin of error.
[tex]CI=(p-E,p+E)\\CI=(0.88-0.0120,0.88+0.0120)\\CI=(0.868,0.892)[/tex]
So, the [tex]90[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern is approximately ([tex]0.868, 0.892[/tex]).
d. Confidence Interval (95%): To calculate the 95% confidence interval, we use the same formula but with a different z-score. For a 95% confidence level, the z-score is approximately 1.96.
[tex]E=1.96*\sqrt{0.88(1-0.88)/ 2000} \\E=1.96*\sqrt{0.0000528} \\E=1.96*0.00727\\E=0.01427\\CI=(0.88-0.0143,0.88+0.0143)\\CI=(0.8657,0.8943)\\[/tex]
So, the [tex]95[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern is approximately ([tex]0.8657, 0.8943[/tex]).
COMPLETE QUESTION:
The Consumer Reports National Research Center conducted a telephone survey of [tex]2000[/tex] adults to learn about the major economic concerns for the future (Consumer Reports, January [tex]2009[/tex]). The survey results showed that [tex]1760[/tex] of the respondents think the future health of Social Security is a major economic concern.
a. What is the point estimate of the population proportion of adults who think the future health of Social Security is a major economic concern (to 2 decimals)?
b. At [tex]90[/tex]% confidence, what is the margin of error (to 4 decimals)?
c. Develop a [tex]90[/tex]% confidence interval for the population proportion of adults who think the future health of Social Security is a major economic concern (to 3 decimals).(, )
d. Develop a [tex]95[/tex]% confidence interval for this population proportion (to 4 decimals).
The probability is 0.271 that the gestation period of a woman will exceed 9 months. In 3000 human gestation periods, roughly how many will exceed 9 months?
Answer:
813 will exceed 9 months.
Step-by-step explanation:
For each women, there are only two possible outcomes. Either they will exceed the gestation period, or they will not. So we use the binomial probability distribution to solve this problem.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
In this problem, we have that:
[tex]n = 3000, p = 0.271[/tex]
In 3000 human gestation periods, roughly how many will exceed 9 months?
[tex]E(X) = np = 3000*0.271 = 813[/tex]
813 will exceed 9 months.
The gestation period should be exceed 9 month is 813.
Given that,
The probability is 0.271 that the gestation period of a woman will exceed 9 months. And, there is 3000 human gestation periodsBased on the above information, the calculation is as follows:
[tex]= 0.271 \times 3,000[/tex]
= 813
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Risk taking is an important part of investing. In order to make suitable investment decisions on behalf of their customers, portfolio managers give a questionnaire to new customers to measure their desire to take financial risks. The scores on the questionnaire are approximately normally distributed with a mean of 49.5 and a standard deviation of 15. The customers with scores in the bottom 10% are described as "risk averse." What is the questionnaire score that separates customers who are considered risk averse from those who are not? Carry your intermediate computations to at least four decimal places. Round your answer to one decimal place.
Final answer:
To identify the score separating risk-averse customers from others in a normally distributed set of questionnaire scores, we find the 10th percentile, which corresponds to a z-score of -1.2816. Using the mean of 49.5 and a standard deviation of 15, we calculate the cutoff score as 30.3.
Explanation:
To find the score that separates the customers who are considered risk averse from those who are not, we must look for the score at the 10th percentile in the normal distribution. Since the scores are approximately normally distributed, we can use the standard z-score table or a statistical calculator to find this value.
The mean (μ) of the distribution is 49.5, and the standard deviation (σ) is 15. We want to find the z-score that corresponds to the cumulative probability of 0.10. Looking at the z-score table or using a calculator, we find that the z-score associated with the bottom 10% of the distribution is approximately -1.2816.
Now we can use the z-score formula:
z = (X - μ) / σ
Where X is the score we are looking for. Substituting the values we have, we get:
-1.2816 = (X - 49.5) / 15
Solving for X:
X = -1.2816 * 15 + 49.5
X ≈ -19.224 + 49.5
X ≈ 30.276
When rounded to one decimal place, we get X = 30.3. Therefore, a score of 30.3 on the questionnaire separates those who are considered risk averse from those who are not.
The vector w=ai+bj is perpendicular to the line ax+by=c and parallel to the line bx−ay=c. It is also true that the acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors that are either normal to the lines or parallel to the lines. Use this information to find the acute angle between the lines below. yx+9y=0, −4x+5y=3
Answer with Step-by-step explanation:
We are given that
[tex]yx+9y=0[/tex]
[tex]-4x+5y=3[/tex]
We have to find the angle between the lines.
[tex]y(x+9)=0[/tex]
[tex]y=0,x+9=0\implies x=-9[/tex]
[tex]y=0[/tex]..(1)
[tex]x=-9[/tex]..(2)
[tex]-4x+5y=3[/tex]..(3)
The angle between two lines
[tex]a_1x+b_1y+c_1=0[/tex]
[tex]a_2x+b_2y+c_2=0[/tex]
[tex]tan\theta=\mid \frac{a_1b_2-b_1a_2}{a_1a_2+b_1b_2}\mid[/tex]
By using the formula the angle between equation (1) and equation (2) is given by
[tex]tan\theta_1=\mid\frac{0\times 0-1\times 1}{0+0}\mid=\infty=90^{\circ}[/tex]degree
[tex]tan90^{\circ}=\infty[/tex]
It is not possible because we are given that the acute angle between intersecting lines that do not cross at right angles is same as the angle determined by vectors that either normal to the lines or parallel to lines.
By using the formula the angle between equation (2) and equation(3)
[tex]tan\theta_2=\mid\frac{1(5)-0(4)}{-4(1)+5(0)}\mid=\frac{5}{4}[/tex]
[tex]\theta_2=tan^{-1}(1.25)[/tex] degree
By using the formula the angle between equation (3) and equation(1)
[tex]tan\theta_3=\mid\frac{-4(1)-5(0)}{-4(0)+5(1)}\mid=\frac{4}{5}[/tex]
[tex]\theta_3=tan^{-1}(\frac{4}{5})[/tex]degree
The following two functions have a common input, year t: R gives the average price, in dollars, of a gallon of regular unleaded gasoline, and P gives the purchasing power of the dollar as measured by consumer prices based on 2010 dollars (a) Using function notation, show how to combine the two functions to create a new function giving the price of gasoline in constant 2010 dollars. 2010 dollars (RP)(t) dollars (at pump) dollars Rit dollars gallon + P(t) gallon gallon gallon O R(t) dollars - P(t) 2010 dolars(R P)(t) 2010 dollars gallon dollars (at pump) dollars (at pump) R(t) gallon · dollars (at pump) R(t)--+ P() dollars (at pump) A(t) dollars . Pit), 2010 dollars 2010 dollars 2010 dollars _- (R- P)(t) dollars gallon 2010 dollars dollars 2010 dollars _- (R + P)(t) gallon (b) What are the output units of the new function? 2010 dollars per gallon gallons per 2010 dollar dollars per gallon 2010 dollars per dollar (at pump) gallons per dollar
Answer:
a) F(t) = R[P(t)]
b) the output units of the new function = F(t) in dollars per gallon
Step-by-step explanation:
a) There are two function R(t) which shows the average price in dollars of a gallon of regular unleaded gasoline and P(t) which shows the purchasing power of the dollar as measured by consumer prices based on 2010 dollars.
To write the function which gives the rice of gasoline in constant 2010 dollars ;
From the analysis , this is an example of a composition of function as such the relationship =
F(t) = R[P(t)]
b) the output units of the new function = F(t) in dollars per gallon
This shows that the value of F(t) is the dependent variable
Final answer:
To determine the real price of gasoline in constant 2010 dollars, combine the functions R(t) and P(t) using the formula R(t) ÷ P(t). This calculation adjusts the nominal price of gasoline for inflation, resulting in the price of gasoline in terms of 2010 dollars, with the output units being 2010 dollars per gallon.
Explanation:
To combine the two functions representing the average price of a gallon of regular unleaded gasoline, R(t), and the purchasing power of the dollar as measured by consumer prices based on 2010 dollars, P(t), into a new function giving the price of gasoline in constant 2010 dollars, we use the formula:
R(t) ÷ P(t)
This formula represents the price of gasoline adjusted for inflation, giving us the real price of gasoline in terms of 2010 dollars. Here, R(t) gives the average price of gasoline in year t, and P(t) gives the purchasing power of the dollar in year t, compared to 2010 dollars. By dividing R(t) by P(t), we adjust the nominal price of gasoline to reflect its real value, accounting for changes in the purchasing power of the dollar over time.
The output units of this new function would be 2010 dollars per gallon. This metric allows economists and analysts to compare the price of gasoline across different years on a level playing field, eliminating the effects of inflation.
Mary's 25th birthday is today, and she hopes to retire on her 65th birthday. She has determined that she will need to have $1,000,000 in her retirement savings account in order to live comfortably. Mary currently has no retirement savings, and her investments will earn 6% annually. How much must she deposit into her account at the end of each of the next 40 years to meet her retirement savings goal
Answer:
I think 25000
Step-by-step explanation:
A wallet contains five $10 bills, three $5 bills, six $1 bills, and no larger denominations. If bills are randomly selected one-by-one from the wallet, what is the probability that at least two bills must be selected to obtain the first $10 bill?
Final answer:
The probability that at least two bills must be selected to obtain the first $10 bill is approximately 24.7%.
Explanation:
To find the probability that at least two bills must be selected to obtain the first $10 bill, we need to calculate the probability of not drawing a $10 bill on the first draw and then drawing a $10 bill on the second draw. In total, there are 5 + 3 + 6 = 14 bills in the wallet.
On the first draw, the probability of not getting a $10 bill is the number of non-$10 bills over the total number of bills, which is (3 $5 bills + 6 $1 bills) / 14 total bills = 9/14.
Assuming a non-$10 bill was drawn first, there are now 13 bills left in the wallet. The probability of drawing a $10 bill on the second draw is now the number of $10 bills remaining over the total number of bills left, which is 5/13.
The combined probability of these two events happening in sequence (not drawing a $10 bill first and then drawing a $10 bill) is the product of their probabilities: (9/14) * (5/13).
Thus, the total probability is (9/14) * (5/13) = 45/182, which simplifies to approximately 0.247 or 24.7%.
An business development executive travels extensively for business. Her company offers two options to offset her driving expenses. Option 1 provides a car allowance of 510 dollars per month and a mileage reimbursement of $0.38/mile for fuel, insurance, and maintenance costs. Option 2 provides a mileage reimbursement of $0.65/mile to cover all expenses associated with owning a car.How many miles would she have to drive each YEAR for the two options to be of equal value. Express your answer in miles to the nearest whole mile.
Answer: 26667 miles
Step-by-step explanation:
According to the statement,
In option 1 we have 510 dollars per month plus $0.38/mile.
In option 2 we have $0.65/mile
Let x be the number of miles.
For a whole year, the option 1 is 510*12+ 0.38 x
For a whole year, the option 2 is 0.65 x
Equating both, we get
6120 + 0.38 x = 0.65 x
Solving, we get
x= 6120/ 0.27
x= 22666.67
x= 26667 miles
We roll two fair 6-sided dice, A and B. Each one of the 36 possible outcomes is assumed to be equally likely. 1) Find the probability that dice A is larger than dice B. 2) Given that the roll resulted in a sum of 5 or less, find the conditional probability that the two dice were equal. 3) Given that the two dice land on different numbers, find the conditional probability that the two dice differed by 2.
Answer:
1) 41.67% probability that dice A is larger than dice B.
2) Given hat the roll resulted in a sum of 5 or less, there is a 20% conditional probability that the two dice were equal.
3) Given that the two dice land on different numbers there is a 26.67% conditional probability that the two dice differed by 2.
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
In this problem, we have these possible outcomes:
Format(Dice A, Dice B)
(1,1), (1,2), (1,3), (1,4), (1,5),(1,6)
(2,1), (2,2), (2,3), (2,4), (2,5),(2,6)
(3,1), (3,2), (3,3), (3,4), (3,5),(3,6)
(4,1), (4,2), (4,3), (4,4), (4,5),(4,6)
(5,1), (5,2), (5,3), (5,4), (5,5),(5,6)
(6,1), (6,2), (6,3), (6,4), (6,5),(6,6)
There are 36 possible outcomes.
1) Find the probability that dice A is larger than dice B.
Desired outcomes:
(2,1)
(3,1), (3,2)
(4,1), (4,2), (4,3)
(5,1), (5,2), (5,3), (5,4)
(6,1), (6,2), (6,3), (6,4), (6,5)
There are 15 outcomes in which dice A is larger than dice B.
There are 36 total outcomes.
So there is a 15/36 = 0.4167 = 41.67% probability that dice A is larger than dice B.
2) Given that the roll resulted in a sum of 5 or less, find the conditional probability that the two dice were equal.
Desired outcomes:
Sum of 5 or less and equal
(1,1), (2,2)
There are 2 desired outcomes
Total outcomes:
Sum of 5 or less
(1,1), (1,2), (1,3), (1,4)
(2,1), (2,2), (2,3)
(3,1), (3,2)
(4,1)
There are 10 total outcomes.
So given hat the roll resulted in a sum of 5 or less, there is a 2/10 = 20% conditional probability that the two dice were equal.
3) Given that the two dice land on different numbers, find the conditional probability that the two dice differed by 2.
Desired outcomes
Differed by 2
(1,3), (2,4), (3,1), (3,5),(4,2),(4,6), (5,3), (6,4).
There are 8 total outcomes in which the dices differ by 2.
Total outcomes:
There are 30 outcomes in which the two dice land of different numbers.
So given that the two dice land on different numbers there is a 8/30 = 0.2667 = 26.67% conditional probability that the two dice differed by 2.
A circle has the center of (1,-5) and a radius of 5 determine the location of the point (4,-1)
"determine the location" or namely, is it inside the circle, outside the circle, or right ON the circle?
well, we know the center is at (1,-5) and it has a radius of 5, so the distance from the center to any point on the circle will just be 5, now if (4,-1) is less than that away, is inside, if more than that is outiside and if it's exactly 5 is right ON the circle.
well, we can check by simply getting the distance from the center to the point (4,-1).
[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ \stackrel{center}{(\stackrel{x_1}{1}~,~\stackrel{y_1}{-5})}\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{-1})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d = \sqrt{[4-1]^2+[-1-(-5)]^2}\implies d=\sqrt{(4-1)^2+(-1+5)^2} \\\\\\ d = \sqrt{3^2+4^2}\implies d =\sqrt{9+16}\implies d=\sqrt{25}\implies \stackrel{\textit{right on the circle}}{d = 5}[/tex]
Suppose the demand for a certain item is given by:
D(p) = - 5p^2-6p+400, where p represents the price of the item in dollars.
a. Find the rate of change of demand with respect to price.
b. Find and interpret the rate of change of demand when the price is $9.
Answer:
a. D'(p) = -10p - 6
b. There is a decrease of 96 units of demand for each dollar increase
Step-by-step explanation:
The demand function is:
[tex]D(p) = - 5p^2-6p+400[/tex]
(a) The derivate of the demand function with respect to price gives us the rate of change of demand:
[tex]\frac{dD(p)}{dp}=D'(p) = -10p-6[/tex]
(b) When p = $9, the rate of change of demand is:
[tex]D'(9) = -10*9-6\\D'(9) = -96\ \frac{units}{\$}[/tex]
This means that, when p = $9, there is a decrease of 96 units of demand for each dollar increase.
Final answer:
The rate of change of demand with respect to price is given by the derivative D'(p) = -10p - 6. When the price is $9, the rate of change of demand is -96, indicating that for each dollar increase in price, demand decreases by 96 items.
Explanation:
To address the demand model problem, we first need to calculate the rate of change of demand with respect to price. This involves taking the derivative of the demand function D(p) = -5p^2 - 6p + 400 with respect to price p. The derivative, D'(p), is calculated as follows:
Differentiate each term with respect to p:
The derivative of -5p^2 is -10p.
The derivative of -6p is -6.
The derivative of a constant (400) is 0.
Combine these to get the rate of change formula D'(p) = -10p - 6.
To find the rate of change of demand when the price is $9, we substitute p with 9 into the rate of change formula:
D'(9) = -10(9) - 6 = -90 - 6 = -96
The rate of change of demand at a price of $9 is -96 items per dollar. This means that for each one dollar increase in price, the quantity demanded decreases by 96 items.
A normal deck of cards has 52 cards, consisting of 13 each of four suits: spades, hearts, diamonds, and clubs. Hearts and diamonds are red, while spades and clubs are black. Each suit has an ace, nine cards numbered 2 through 10, and three "face cards." The face cards are a jack, a queen, and a king. Answer the following questions for a single card drawn at random from a well-shuffled deck of cards. a. What is the probability of drawing a king of any suit? b. What is the probability of drawing a face card that is also a spade? c. What is the probability of drawing a card without a number on it? d. What is the probability of drawing a red card? What is the probability of drawing an ace? What is the probability of drawing a red ace? Arc these events ("ace" and "red") mutually exclusive? Are they independent? List two events that are mutually exclusive
Final answer:
a. The probability of drawing a king of any suit is 1/13. b. The probability of drawing a face card that is also a spade is 3/26. c. The probability of drawing a card without a number is 4/13. d. The probability of drawing a red card is 1/2. The probability of drawing an ace is 1/13. The probability of drawing a red ace is 1/26.
Explanation:
a. There are 4 kings in a deck of cards, one for each suit. So, the probability of drawing a king of any suit is the number of king cards divided by the total number of cards in the deck: 4/52 = 1/13 = 0.077 or 7.7%.
b. There are 3 face cards in each suit, and there are 2 black suits (spades and clubs). So, the probability of drawing a face card that is also a spade is the number of face cards (3) multiplied by the number of black suits (2), divided by the total number of cards in the deck: (3 * 2)/52 = 6/52 = 3/26 = 0.115 or 11.5%.
c. A card without a number refers to a face card (jack, queen, or king) or an ace. There are 12 face cards and 4 aces in a deck. So, the probability of drawing a card without a number is the number of face cards plus the number of aces divided by the total number of cards in the deck: (12 + 4)/52 = 16/52 = 4/13 = 0.308 or 30.8%.
d. There are 26 red cards in a deck (hearts and diamonds) and 52 total cards. So, the probability of drawing a red card is the number of red cards divided by the total number of cards: 26/52 = 1/2 = 0.5 or 50%. The probability of drawing an ace is 4/52 = 1/13 = 0.077 or 7.7%. The probability of drawing a red ace is the number of red aces divided by the total number of cards: 2/52 = 1/26 = 0.038 or 3.8%.
These events are mutually exclusive because a card cannot be an ace and also be a non-ace card at the same time. However, they are not independent because the probability of drawing a red ace would change if an ace had already been drawn.
Two events that are mutually exclusive are drawing a spade and drawing a heart. You cannot draw a card that is both a spade and a heart at the same time.
An author collected the times (in minutes) it took him to run 4 miles on various courses during a 10-year period. The accompanying histogram shows the times. Describe the distribution and summarize the important features. What is it about running that might account for the shape of the histogram?
Answer:
By looking at the histogram, we can conclude that the distribution is unimodal and skewed to the right. The modal value lies around 30 to 31 minutes and most of the running times range between 29 and 32 minutes.
The histogram is skewed to the right with most of the outliers present at the higher running times because practically, it is most likely possible for a person to run slow and take more time to run 4 miles rather than run fast and take less time.
Step-by-step explanation:
We say that the distribution is unimodal because there is only one peak which is the highest.
The distribution is skewed to the right because most of the outliers are present at the left side of the peak.
What is the difference between the population and sample regression functions? Is this a distinction without difference?
Answer:
See explanation below.
Step-by-step explanation:
When we want to fit a linear model given by:
[tex] y = \beta_0 + \beta_1 x[/tex]
Where y is a vector with the observations of the dependent variable, [tex]\beta_0 , \beta_1 [/tex] the parameters of the model and x the vector with the observations of the independent variable.
For this case this population regression function represent the conditional mean of the variable Y with values of X constant. And since is a population regression the parameters are not known, for this reason we use the sample data to obtain the sample regression in order to estimate the parameters of interest [tex] \beta_0, \beta_1[/tex]
We can use any method in order to estimate the parameters for example least squares minimizing the difference between the fitted and the real observations for the dependenet variable. After we find the estimators for the regression model then we have a model with the estimated parameters like this one:
[tex] \hat y = \hat b_0 +\hat b_1 x[/tex]
With [tex] \hat \beta_0 = b_o , \hat \beta_1 = b_1[/tex]
And this model represent the sample regression function, and this equation shows to use the estimated relation between the dependent and the independent variable.
A marijuana survey included 1610 responses from a list of approximately 241,500,000 adults 10) in the U.S. from which every 150.000 name was surveyed. Identify which of these types of sampling is used: A) Stratified B) Cluster C) ConvenienceD) Systematic E) Simple random
Answer:
the Fact that 1610 responses where gotten from the original population of
241 500 000 makes this a convenience sampling.
Step-by-step explanation:
convenience Sampling : this is a type of non-probability sampling that involves the sample being drawn from that part of the population that is close to hand.
Consider the two different numbers 327b (327 is base b) and 327b 1 (327 in base b 1), where b is a positive integer 8 or greater. If the difference between these two numbers is 89, what is the value of b
Answer:
Value of b = 14
Step-by-step explanation:
The detailed calculations with steps is shown in the attachment.
A 40% antifreeze solution is to be mixed with a 70% antifreeze
solution to get 240 liters of a 50% solution. How many liters of the
40% solution and how many liters of the 70% solution will be used?
Answer: 160 liters of the
40% solution and 80 liters of the 70% solution will be used.
Step-by-step explanation:
Let x represent the number of liters of 40% antifreeze solution that should be used.
Let y represent the number of liters of 70% antifreeze solution that should be used.
The volume of the mixture to be mixed is 240 liters. It means that
x + y = 240
The 40% antifreeze solution is to be mixed with a 70% antifreeze
solution to get 240 liters of a 50% solution. This means that
0.4x + 0.7y = 0.5(240)
0.4x + 0.7y = 120 - - - - - - - - - - - -1
Substituting x = 240 - y into equation 1, it becomes
0.4(240 - y) + 0.7y = 120
96 - 0.4y + 0.7y = 120
- 0.4y + 0.7y = 120 - 96
0.3y = 24
y = 24/0.3
y = 80
x = 240 - y = 240 - 80
x = 160
[tex]160 \ litres[/tex] of [tex]40 \%[/tex] antifreeze solution and [tex]80 \ litres[/tex] of [tex]70 \%[/tex] antifreeze solutions will be used.
Given, two solutions namely [tex]40 \%[/tex] antifreeze and [tex]70 \%[/tex] antifreeze solutions.
Let [tex]x[/tex] litres of the [tex]40 \%[/tex] antifreeze solution and [tex]y[/tex] litres of the [tex]70 \%[/tex] antifreeze solutions will be used.
Total volume of the solution,
[tex]x+y=240..........(1)[/tex]
Now, [tex]40\%[/tex] of antifreeze solution is to be mixed with a [tex]70 \%[/tex] of antifreeze
solution to get 240 liters of a [tex]50 \%[/tex] solution,
[tex]0.4x+0.7y=240\times 0.5[/tex]
[tex]0.4x+0.7y=120........(2)[/tex]
From Equation (1) [tex]y=240-x[/tex], substitute the value of [tex]y[/tex] in Equation (2),we get
[tex]0.4x+0.7(240-x)=120\\0.4x+168-0.7x=120\\-0.3x=120-168\\-0.3x=-48\\x=160[/tex]
Putting the value of [tex]x=160[/tex], we get
[tex]y=240-160[/tex]
[tex]y=80[/tex].
Hence [tex]160 \ litres[/tex] of [tex]40 \%[/tex] antifreeze solution and [tex]80 \ litres[/tex] of [tex]70 \%[/tex] antifreeze solutions will be used.
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PLEASE HELP 50 COINS!!!!
Answer:
Therefore the measurement of EF,
[tex]EF=1.98\ units[/tex]
Step-by-step explanation:
Given:
In Right Angle Triangle DEF,
m∠E=90°
m∠D=26°
∴sin 26 ≈ 0.44
DF = Hypotenuse = 4.5
To Find:
EF = ? (Opposite Side to angle D)
Solution:
In Right Angle Triangle DEF, Sine Identity,
[tex]\sin D= \dfrac{\textrm{side opposite to angle D}}{Hypotenuse}\\[/tex]
Substituting the values we get
[tex]\sin 26= \dfrac{EF}{DF}=\dfrac{EF}{4.5}[/tex]
Also, sin 26 ≈ 0.44 .....Given
[tex]EF = 4.5\times 0.44=1.98\ units[/tex]
Therefore the measurement of EF,
[tex]EF=1.98\ units[/tex]
An experimenter is studying the effects of temperature, pressure, and different type of catalysts.
If there are 3 different temperatures, 4 different pressures, and 5 different catalysts, how many experimental runs are available?
Answer: 60
Step-by-step explanation:
Given : The number of choices for different temperatures = 3
The number of choices for different pressures = 4
The number of choices for different catalyst = 5
Since , the experimenter is studying the effects of temperature, pressure, and different type of catalysts.
Then by using the Fundamental principle of counting , we have
The number of experimental runs are available = (number of choices for temperatures ) x (number of choices for pressures) x( number of choices for catalyst)
= 3 x 4 x 5 = 60
Hence, the number of experimental runs are available = 60
Strain-displacement relationship) Consider a unit cube of a solid occupying the region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 After loads are applied, the displacements are
Answer:
please see answers are as in the explanation.
Step-by-step explanation:
As from the data of complete question,
[tex]0\leq x\leq 1\\0\leq y\leq 1\\0\leq z\leq 1\\u= \alpha x\\v=\beta y\\w=0[/tex]
The question also has 3 parts given as
Part a: Sketch the deformed shape for α=0.03, β=-0.01 .
Solution
As w is 0 so the deflection is only in the x and y plane and thus can be sketched in xy plane.
the new points are calculated as follows
Point A(x=0,y=0)
Point A'(x+αx,y+βy)
Point A'(0+(0.03)(0),0+(-0.01)(0))
Point A'(0,0)
Point B(x=1,y=0)
Point B'(x+αx,y+βy)
Point B'(1+(0.03)(1),0+(-0.01)(0))
Point B'(1.03,0)
Point C(x=1,y=1)
Point C'(x+αx,y+βy)
Point C'(1+(0.03)(1),1+(-0.01)(1))
Point C'(1.03,0.99)
Point D(x=0,y=1)
Point D'(x+αx,y+βy)
Point D'(0+(0.03)(0),1+(-0.01)(1))
Point D'(0,0.99)
So the new points are A'(0,0), B'(1.03,0), C'(1.03,0.99) and D'(0,0.99)
The plot is attached with the solution.
Part b: Calculate the six strain components.
Solution
Normal Strain Components
[tex]\epsilon_{xx}=\frac{\partial u}{\partial x}=\frac{\partial (\alpha x)}{\partial x}=\alpha =0.03\\\epsilon_{yy}=\frac{\partial v}{\partial y}=\frac{\partial ( \beta y)}{\partial y}=\beta =-0.01\\\epsilon_{zz}=\frac{\partial w}{\partial z}=\frac{\partial (0)}{\partial z}=0\\[/tex]
Shear Strain Components
[tex]\gamma_{xy}=\gamma_{yx}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=0\\\gamma_{xz}=\gamma_{zx}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}=0\\\gamma_{yz}=\gamma_{zy}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}=0[/tex]
Part c: Find the volume change
[tex]\Delta V=(1.03 \times 0.99 \times 1)-(1 \times 1 \times 1)\\\Delta V=(1.0197)-(1)\\\Delta V=0.0197\\[/tex]
Also the change in volume is 0.0197
For the unit cube, the change in terms of strains is given as
[tex]\Delta V={V_0}[(1+\epsilon_{xx})]\times[(1+\epsilon_{yy})]\times [(1+\epsilon_{zz})]-[1 \times 1 \times 1]\\\Delta V={V_0}[1+\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}+\epsilon_{xx}\epsilon_{yy}+\epsilon_{xx}\epsilon_{zz}+\epsilon_{yy}\epsilon_{zz}+\epsilon_{xx}\epsilon_{yy}\epsilon_{zz}-1]\\\Delta V={V_0}[\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\[/tex]
As the strain values are small second and higher order values are ignored so
[tex]\Delta V\approx {V_0}[\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\ \Delta V\approx [\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\[/tex]
As the initial volume of cube is unitary so this result can be proved.
The formula d = 6 t − 11 d=6t-11 expresses a car's distance (in feet) from a stop sign, d d, in terms of the number of seconds t t since it started moving. Determine the car's average speed over each of the following intervals of time.a. From t=3 to t=6 seconds...
b. From t=6 to t=6.5 seconds...
c. From t=6.5 to t=7 seconds...
Answer:
a) 6feet/secs
b) 6feet/secs
c) 6feet/secs
Step-by-step explanation:
The detailed steps are as shown in the attachment
The average speed of the car in each time interval is calculated by first evaluating the distance formula at the endpoints of the interval, subtracting to find the distance travelled, and then dividing by the time taken to travel that distance.
Explanation:The given formula is
d = 6t - 11
, where 'd' is the distance in feet, and 't' is the time in seconds since the car started moving. Firstly, to find the average speed, which is the distance travelled divided by time taken, we need to calculate the distance travelled in each interval. For instance, for the interval from 't=3' to 't=6', we first calculate the distances 'd' at t=3 and t=6 by substituting them into the equation, then subtracting the two to get the distance travelled over this time interval. Similarly, the distances travelled in the intervals from t=6 to t=6.5 seconds and t=6.5 to t=7 seconds were calculated. Finally, the
average speed
in each time interval is obtained by dividing that interval's travelled distance by the time taken.
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A sales representative must visit nine cities. There are direct air connections between each of the cities. Use the multiplication rule of counting to determine the number of different choices the sales representative has for the order in which to visit the cities.
Answer:
362880
Step-by-step explanation:
Given that a sales representative must visit nine cities. There are direct air connections between each of the cities
Since there are direct connections between any two pairs the sales rep can visit in any order as he wishes.
He has 9 ways to select first city, now remaining cities are 8. He can visit any one in 8 ways.
i.e. No of ways of visiting 9 cities in any order = 9P9
= 9!
= 362880
So no of ways he visits the cities since there are direct connections between any two cities is
362880
This is a permutation problem. The sales representative has 9 factorial (9*8*7*6*5*4*3*2*1 = 362,880) different choices for the order to visit the cities.
Explanation:The subject of this question is a part of combinatorics, specifically Permutations. In this scenario, the sales representative has 9 different cities to visit and the order in which the cities are visited is important.
Using the multiplication rule of counting, the number of ways he can visit these cities is 9 factorial (9!). In general, the abbreviation 'n!' denotes the product of all positive integers less than or equal to n.
So for 9 cities this would be: 9*8*7*6*5*4*3*2*1 = 362,880 different choices for order to visit the cities.
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Someone trips on the sidewalk, droppingan urn containing 3 blue and 3 yellow marbles. themarbles roll away, but come to a stop (all in a row)on a crack in the cement. What is the probabilityof the three blue marbles ending up next to one an-other (i.e., without any yellow marbles in betweenthem)
Answer:
The probability of the three blue marbles ending up next to one another (i.e., without any yellow marbles in between them is 1/5 or 0.2.
Step-by-step explanation:
The 6 marbles can be arranged in 6! ways, But, there are 3 identical marbles of similar colour in two separate cases,
So, 6 marbles, consisting of 3 blue and 3 yellow marbles can be arranged in 6!/(3!3!) ways = 20 ways.
But, to arrange the six marbles in such a way that the 3 blue marbles end up next to one another without any yellow marble between them. This can be done by viewing the 3 blue marbles as one. Therefore, there are 4 marbles; 3 identical, blue marbles and 1 special marble.
To arrange that, it is, 4!/(3!1!) = 4
The probability of the three blue marbles ending up next to one another (i.e., without any yellow marbles in between them will be 4/20 = 1/5 or 0.2.
Final answer:
The probability that the three blue marbles will end up next to each other is 1/120. This is calculated by considering the blue marbles as one unit and arranging them with the yellow marbles, leading to 24 total arrangements, but since the blue marbles are indistinguishable, the number of favorable outcomes is the same as the arrangement of the yellow marbles, which is 6. The total number of possible outcomes is 720, calculated by 6 factorial.
Explanation:
The question asks for the probability that three blue marbles will end up next to each other after being dropped and rolling into a crack. To solve this, consider the three blue marbles as a single unit. Since there are also three yellow marbles, we can arrange four units (three blue marbles together as one unit and the three individual yellow marbles) in 4! (4 factorial) ways, which is equal to 24. However, the three blue marbles as a single unit can also be arranged among themselves in 3! (3 factorial) ways, but since they are indistinguishable, we don't consider these arrangements separate. So, there is only one way to arrange the blue block. Therefore the total number of favorable outcomes is the same as the number of ways to arrange the yellow marbles, which is also 3! or 6. To find the probability, we divide the favorable outcomes by the total possible outcomes without restrictions.
The total possible outcomes without any restrictions can be calculated assuming all 6 marbles are distinct, which gives us 6! (6 factorial) arrangements, equal to 720. Applying the probability formula, we have-
Probability = Favorable outcomes / Total outcomes = 6 (the number of ways to arrange the yellow marbles) / 720 possible arrangements = 6/720 = 1/120.
Therefore, the probability that the three blue marbles will end up next to each other is 1/120.
An experimenter is randomly sampling 4 objects in order from among 43 objects. What is the total number of samples in the sample space?
Final answer:
The total number of samples in the sample space can be found by using the concept of permutations and the rule of product.Therefore total number of samples in the sample space is 352,560.
Explanation:
The total number of samples in the sample space can be found by using the concept of permutations and the rule of product. Since there are 43 objects and we are sampling 4 objects in order, we can use the formula:
nPr = n! / (n - r)!
where n is the total number of objects and r is the number of objects being sampled. Plugging in the values, we get:
43P4 = 43! / (43 - 4)!
Simplifying, we have:
43P4 = 43! / 39!
which can be further simplified to:
43P4 = (43 * 42 * 41 * 40 * 39!) / 39!
The 39! terms cancel out, leaving us with:
43P4 = 43 * 42 * 41 * 40
Evaluating this expression, we find that the total number of samples in the sample space is 352,560.
The total number of samples in the sample space when randomly sampling 4 objects from 43 is calculated using permutations, resulting in 2,906,880 possible samples.
The student has asked about finding the total number of samples in the sample space when randomly sampling 4 objects in order from among 43 objects. This can be calculated using the formula for permutations, given that the order of selection matters and repeats are not allowed. The formula for permutations of n objects taken r at a time is nPr = n! / (n - r)!, where n! denotes the factorial of n, and (n-r)! denotes the factorial of (n-r).
In this case, n = 43 (the total number of objects) and r = 4 (the number of objects being selected). Thus, the calculation will be 43P4 = 43! / (43 - 4)! = 43! / 39!. Carrying out the calculation, 43P4 equals 43 × 42 × 41 × 40, which results in 2,906,880 possible samples.
Given the following sequence, what is the 10th term of the sequence? Assume that the sequences start with an index of 1. Sequence: The nth term is 3.
Answer:
the 10th term of the sequence is 3
Step-by-step explanation:
A sequence is a list of numbers or objects in a specific order
The nth term given in the question is not a function of n
i.e aₙ= 3
Since the sequence starts with an index of 1
a₁=3
All other terms in the sequence will also be 3. Meaning that it is an arithmetic progression with a common difference of 0.
The 10th term is given by
a₁₀= 3
A bacteria culture is initially 10 grams at t=0 hours and grows at a rate proportional to its size. After an hour the bacteria culture weighs 11 grams. At what time will the bacteria have tripled in size?
Answer: It will take 11.56 hours .
Step-by-step explanation:
Exponential growth in population or size formula :
[tex]P(t)=P_0e^{rt}[/tex]
, where [tex]P_0[/tex] = initial size
r= rate of growth
t= time period
As per given , we have
[tex]P_0=10[/tex] grams
At t= 1 , P(t)= 11 grams
Then,
[tex]11=10e^{r(1)}\\\\ 1.1= e^r\\\\\text{Taking natural log on both sides , we get} \\\\\ln (1.1)=r\ln (e)\\\\ r=\ln (1.1)\\\\ r=0.0953101798043\approx0.095[/tex]
When, the bacteria have tripled in size , P(t) = 3 x10 = 30
Then,
[tex]30=10e^{0.095t}\\\\ 3=e^{0.095t}[/tex]
[tex]\text{Taking natural log on both sides , we get}\\\\ \ln 3=0.095t\\\\ t=\dfrac{\ln3}{0.095}\\\\ t=\dfrac{1.09861228867}{0.095}\approx11.56[/tex]
Hence, it will take 11.56 hours .
A bacteria culture is initially 10 grams at t=0 hours & grows at a rate proportional to its size , After an hour the bacteria culture weighs 11 grams , The bacteria takes 11.56 hours to have tripled in size.
To find the time of bacteria when increasing the growth to tripled.
Given : when time=0 hours , weight=10 grams.
when time=1 hours , weight=11 grams.
To find: when time= ? hours , weight=30grams.
Here according to question, initial size = 10 grams we have asked for tripled in size i.e. 30 grams.
Now we knows that,
The formula for exponential growth in population or size is
[tex]\rm (P)=P_0e^{rt}[/tex] where,
[tex]\rm P_0=initial\;size\\\\r= rate\;of\;growth\\\\t= time \;period[/tex]
Now, we put the value in formula we get,
[tex]\rm P_0=10\;grams \\\\when ,\\\;\;t=1\;hour P(t)=11 grams\\Then,\\11=10e^{r(1)\\1.1 =e^r\\\\\rm Taking \;log(natural)\;both\;the\; side \;on \;solving\;we\;get,\\ln(1.1)=r\;ln(e)\\r=ln(1.1)\\r=0.953101798043\approx0.095[/tex]
Now when the bacteria increase its size to triple
[tex]\rm P(t) = 3 \times 10 = 30[/tex]
Then, according to the formula we substitute values in the formula,
[tex]\rm 30=10e^{0.095t}\\\\3=e^{0.095t}\\\\Again \;we \;take\;natural\;log\;on \;both\;the\;sides, we\;get\\ln\;3=0.095t\\\\t=\dfrac{\rm ln\;3}{0.095}\\\\\\\\\rm t= \dfrac{1.09861228867}{0.095} \\\\\ t=approx \; 11.56[/tex]
Therefore, The bacteria takes 11.56 hours to have tripled in size.
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A manufacturer of jeans has plants in California, Arizona, and Texas. A group of 25 pairs of jeans is randomly selected from the computerized database, and the state in which each is produced is recorded.
CA AZ AZ TX CA
CA CA TX TX TX
AZ AZ CA AZ TX
CA AZ TX TX TX
CA AZ AZ CA CA
a. What is the experimental unit?
b. What is the variable being measured? Is it qualitative or quantitative?
c. Construct a pie chart to describe the data.
d. Construct a bar chart to describe the data.
e. What proportion of the jeans are made in Texas?
f. What state produced the most jeans in the group?
g. If you want to find out whether the three plants produced equal number of jeans, or whether one produced more jeans that the others, how can you use the charts from parts c and d to help you? What conclusions can you draw from these data?
Answer and Step-by-step explanation:
a) In statistics, an experimental unit is one member of a set of entities being studied. Experimental units are the individuals on whom an experiment is being performed on.
25 pairs of jeans are randomly selected, hence, a single experimental unit is a pair of jeans.
b) Variables are the qualities/topics being investigated. Qualitative variables puts the variables in categories while quantitative variables involve numerical variables.
This question focuses on which state each pair of jeans is being produced, therefore this quality categorizes the jeans according to which state they were produced in. Hence, the variable being measured is a qualitative one.
c) For the pie chart, we need the frequency of the pair's of the jeans according to which state they were produced in.
California, CA, frequency = 9
Arizona, AZ, frequency = 8
Texas, TX, frequency = 8
Total = 25
A pie chart is based on 360°
CA will occupy (9/25) × 360° = 129.6°
AZ will occupy (8/25) × 360° = 115.2°
TX will occupy (8/25) × 360° = 115.2°
The pie chart is drawn with Microsoft Excel and presented in the image attached to this answer.
d) The bar chart is drawn with Microsoft Excel and presented in the image attached to this question. Each bar has equal width, but the height of each bar corresponds to its frequency.
e) The proportion of jeans produced in Texas = 8/25 = 0.32
f) The state that produces the highest number of jeans is the one with the highest frequency. That is California with frequency of 9 out of 25.
(g) The plant that produced more jeans than the others is the state in the
pie chart of part (c) with the largest angle (129.6°) or slice (California) and is the state in the bar graph of part (d) that has the highest bar.
From the data, it is evident that the three states make almost the same number of jeans, but California slightly edges the other two by being the state that produces the most number of jeans. Arizona and Texas produce the similar amounts of Jeans.
Although, this is just a sample out of a whole large quantity of Jeans, the laws of statistics and probability makes this random selection a representative of the whole set of jeans produced.
Hope this helps!
In this case, the experimental unit is the pair of jeans, the variable measured is the production state which is qualitative. The proportional production and most productive state can be determined by counting entries, and pie and bar charts can illustrate production distribution.
Explanation:a. The experimental unit is the individual pair of jeans.
b. The variable being measured is the state in which the jeans are produced, which is a qualitative variable.
c. and d. To construct the pie and bar charts, you simply need to count the total number of jeans produced in each state and represent this on the charts. This can be useful for visualizing the distribution of production across the states.
e. The proportion of jeans made in Texas can be found by counting the number of 'TX' entries and dividing by the total number of jeans (25).
f. The state that produced the most jeans is the one with the highest count in your data set. You would need to tally the appearances of each state to determine this.
g. The charts from parts c and d can assist you in determining whether there is a noticeable difference in production between the states. If one section of the pie chart or one bar on the bar graph is significantly larger than the others, it would suggest that state produces significantly more jeans. Similarly, if all sections/bars are similar in size, it suggests equal production across the states.
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