Answer:
a) [tex] \bar X = 52[/tex]
[tex] Median = 55[/tex]
b) [tex]Q_1= \frac{44+44}{2}=44[/tex]
[tex]Q_3= \frac{57+60}{2}=58.5[/tex]
c) [tex] Range = Max -Min = 69-36=33[/tex]
d) [tex] s^2 =100.1429[/tex]
[tex] s= \sqrt{100.143}=10.0071[/tex]
e) [tex] Lower = Q_1 -1.5 IQR = 44-1.5(58.5-44) = 22.25[/tex]
[tex] Upper = Q_1 +1.5 IQR = 44+1.5(58.5-44) = 65.75[/tex]
A possible outlier would be the value of 69 since its above the upper limti for the boxplot.
Step-by-step explanation:
For this case we have the following dataset:
55 56 44 43 44 56 60 62 57 45 36 38 50 69 65
A total of 15 observations
Part a
We calculate the mean with the following formula:
[tex] \bar X = \frac{\sum_{i=1}^{15} X_i}{15}[/tex]
And for this case we got [tex] \bar X = 52[/tex]
For the median we ust need to order the data on increasing way like this:
36, 38,43,44,44,45,50,55,56,56,57,60,62,65,69
Since the number of observations is an odd number the median would be on the 8 position from the dataset ordered on this case:
[tex] Median = 55[/tex]
Part b
In order to calculate the Q1 we need to select the following data:
36, 38,43,44,44,45,50,55
And the Q1 would be the average between the 4 and 5 positions like this:
[tex]Q_1= \frac{44+44}{2}=44[/tex]
And for the Q3 we select these values:
55,56,56,57,60,62,65,69
And the Q3 would be the average between the 4 and 5 positions like this:
[tex]Q_3= \frac{57+60}{2}=58.5[/tex]
Part c
The Range is defined as:
[tex] Range = Max -Min = 69-36=33[/tex]
Part d
In order to calculate the sample variance we can use the following formula:
[tex] s^2 = \frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}[/tex]
And if we replace we got:
[tex] s^2 =100.1429[/tex]
And the deviation is just the square root of the variance:
[tex] s= \sqrt{100.143}=10.0071[/tex]
Part e
For this case we need to find the lower and upper limits for the boxplot given by:
[tex] Lower = Q_1 -1.5 IQR = 44-1.5(58.5-44) = 22.25[/tex]
[tex] Upper = Q_1 +1.5 IQR = 44+1.5(58.5-44) = 65.75[/tex]
A possible outlier would be the value of 69 since its above the upper limti for the boxplot.
gasoline wholesale distributor has bulk storage tanks holding a fixed supply. The tanks are filled every Monday. Of interest to the wholesaler is the proportion of this supply that is sold during the week. Over many weeks, this proportion has been observed to be modeled fairly well by a beta distribution with alpha = 4 and beta = 2. Find the probability that at least 90% of the stock will be sold in a given week? a. 0.07 b. 0.05 c. 0.09 d. 0.06 e. 0.08
Answer:
e. 0.08
Step-by-step explanation:
In the question above, a certain quantity of goods was supplied while a specific quantity of goods was sold per week. In a given week, if the number of proportion sold is X, therefore:
f(x) = {Γ(4+2)/Γ(4)Γ(2) x^3 (1-x), 0≤x≤1 ; 0, elsewhere
and
P(X greater than 0.9) = [tex]\int\limits^1_ {0.9} \, 20(x^{3} - x^{4}) dx[/tex] = 20*{(y^4/4)[1,0.9] - (y^5/5)[1,0.9]} = 20*{(0.25 - 0.164) - (0.20 - 0.118)} = 20*{0.086 - 0.0819} = 20*0.0041 = 0.082
Therefore the probability of the proportion sold is approximately 0.082
The probability that at least 90% of the stock will be sold in a given week is approximately 0.05. The correct answer option is b. 0.05
Here's how to calculate it:
1. Given the beta distribution with parameters alpha = 4 and beta = 2, we want to find [tex]\( P(X > 0.9) \)[/tex], where X represents the proportion of stock sold in a week.
2. Since [tex]\( P(X > 0.9) = 1 - P(X \leq 0.9) \)[/tex], we need to find the cumulative distribution function (CDF) of the beta distribution and then subtract it from 1.
3. Using the incomplete beta function formula [tex]\( I_{x}(\alpha, \beta) \)[/tex], we have:
[tex]\[ P(X > 0.9) = 1 - I_{0.9}(4, 2) \][/tex]
4. Calculating [tex]\( I_{0.9}(4, 2) \)[/tex] using software or a calculator gives approximately 0.95.
5. Subtracting this from 1:
[tex]\[ P(X > 0.9) = 1 - 0.95 = 0.05 \][/tex]
So, the correct answer is b. 0.05.
Quota sampling is most commonly used in a. descriptive research. b. collecting primary data. c. surveys. d. population research. e. exploratory studies.
Quota sampling is most commonly used in surveys. It is a non-probability sampling technique where researchers select individuals who meet certain criteria to be included in the sample.
Explanation:Quota sampling is most commonly used in surveys. It is a non-probability sampling technique where researchers select individuals who meet certain criteria to be included in the sample.
In quota sampling, the researcher sets quotas or targets for each subgroup they want to include in the sample based on certain characteristics. For example, if the researcher wants equal representation of males and females in the sample, they would set quotas for each gender and continue selecting individuals until the quotas are met.
Quota sampling is often used when it is not possible or practical to obtain a random sample, but the researcher still wants to ensure representation of different subgroups within the sample.
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Determine whether the relation R defined below is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. For each property, either explain why R has that property or give an example showing why it does not.
a) Let A = {1, 2, 3, 4} and let R = { (2, 3) }
b) Let A = {1, 2, 3, 4} and let R = { (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 4), (3, 1), (3, 3), (4, 1), (4, 4) }.
Answer:
See below
Step-by-step explanation:
Remember some definitions about binary relations. If R⊆S×S then
R is reflexive if (a,a)∈R for all a∈SR is irreflexive if (a,a)∉R for all a∈SR is symmetric if (a,b)∈R implies (b,a)∈R for all a,b∈SR is asymmetric if (a,b)∈R implies (b,a)∉R for all a,b∈SR is antisymmetric if (a,b)∈R and (b,a)∈R imply that a=b, for all a,b∈SR is transitive if (a,b)∈R and (b,c)∈R imply (a,c)∈R for all a,b,c∈Sa) R is not reflexive since (1,1)∉R.
R is irreflexive, since (a,a)∉R for all a=1,2,3,4
R is asymmetric: (2,3)∈R and (3,2)∉R (thus R is not symmetric).
R is antisymmetric, there are no cases to check. R is transitive, there are no cases to check.
b) R is reflexive, checking case by case, (a,a)∈R for all a=1,2,3,4. Hence R is not irreflexive.
R is not asymmetric: (1,2)∈R but (2,1)∈R. R is not symmetric, since (4,1)∈R but (1,4)∉R
R is not antisymmetric: (1,2)∈R and (2,1)∈R but 1≠2.
R is not transitive: (1,2)∈R and (2,4)∈R but (1,4)∉R.
A small business makes 3-speed and 10-speed bicycles at two different factories. Factory A produces 16 3-speed and 20 10-speed bikes in one day while factory B produces 12 3-speed and 20 10- speed bikes daily. It costs $1000/day to operate factory A and $800/day to operate factory B. An order for 80 3-speed bikes and 140 10-speed bikes has just arrived How many days should each factory be operated in order to fill this order at a minimum cost? (Give your answers correct to two decimal places.) Factory A should be operated Factory B should be operated days. days. What is the minimum cost? (Give your answer correct to the nearest dollar.)
Answer:
Factory A should be operated 6.11 days
Factory B should be operated 6.88 days
The minimum cost for factory A is $6,110
The minimum cost for factory B is $5,504
Step-by-step explanation:
Factory A
Daily production: 16 3-speed and 20 10-speed bikes
Total daily production = 16+20 = 36 speed bikes
Order: 80 3-speed and 140 10-speed bikes
Total order = 80+140 = 220 speed bikes
Number of days = 220/36 = 6.11 days
Cost per day = $1,000
Minimum cost for 6.11 days = $1000 × 6.11 = $6,110
Factory B
Daily production: 12 3-speed and 20 10-speed bikes
Total daily production = 12+20 = 32 speed bikes
Total order = 220 speed bikes
Number of days = 220/36 = 6.88 days
Cost per day = $800
Cost for 6.88 days = $800 × 6.88 = $5,504
The business should operate Factory A for about 2.92 days and Factory B for about 4.17 days to fill the order at a minimum cost of $7148.
Explanation:This problem can be solved using Linear Programming, a mathematical model used in optimization problems. Suppose we let A be the number of days factory A operates and B be the number of days factory B operates. The cost to operate the factories is given by the equation C = 1000A + 800B.
The constraints are as follows:
Factory A produces 16 3-speed bikes per day and factory B produces 12, so 16A + 12B ≥ 80. Both factories produce 20 10-speed bikes per day, so 20A + 20B ≥ 140.By solving these equations, you find that Factory A should be operated for about 2.92 days and Factory B should be operated for 4.17 days, giving a minimum cost of about $7148.
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On a test, 74% of the questions are answered correctly. If 111 questions are correct, how many questions are on the test?
Answer:
150
Step-by-step explanation:
You have to divide the correct answers (111) by the total amount of questions (x) in order to find the 74% (0.74)
As per the unitary method, there are 150 questions on the test.
Let's start by breaking down the information we have been given. We know that 74% of the questions on the test were answered correctly, and the number of questions answered correctly is 111.
Step 1: Convert Percentage to Decimal
To make calculations easier, we need to convert the percentage into a decimal. To do this, we divide 74 by 100, which gives us 0.74.
Step 2: Set Up the Equation
Now we can set up an equation to solve for the total number of questions on the test. Let's denote the total number of questions as "x". Since 74% of the questions were answered correctly, we can say that 0.74x questions were answered correctly.
Step 3: Solve for Total Questions
We are given that 111 questions were answered correctly. So, we can set up the equation:
0.74x = 111
Step 4: Solve for "x"
To solve for "x", we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 0.74:
x = 111 / 0.74
x = 150
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60% of all vehicles examined at a certain emissions inspection station pass the inspection. Assuming that successive vehicles pass or fail independently of one another, calculate the probability that at least one of the next three vehicles fail. (Give your answer as a decimal number with 3 digits of precision.)
Answer:
60%
Step-by-step explanation:
The percentage does not change.
Probability at least one of next 3 vehicles fail inspection: approximately 0.784 (to 3 decimal places).
To find the probability that at least one of the next three vehicles fail, we can calculate the probability of the complementary event, i.e., the probability that all three vehicles pass, and then subtract it from 1.
Given that 60% of vehicles pass the inspection, the probability that one vehicle fails the inspection is 1 - 0.60 = 0.40.
Since the vehicles pass or fail independently, the probability that all three vehicles pass is:
[tex]\[0.60 \times 0.60 \times 0.60 = 0.216\][/tex]
Now, the probability that at least one vehicle fails is:
[tex]\[1 - 0.216 = 0.784\][/tex]
So, the probability that at least one of the next three vehicles fail is approximately [tex]\(0.784\)[/tex] (to 3 decimal places).
A recent survey found that 71% of senior adults wear glasses for driving. In a group of 20 senior adults, how like that no more than 10 wear glasses for driving? Group of answer choices 95.2% 4.8% 3.1% 1.7%
Answer:
[tex] P(X\leq 10) = 1- 0.961525= 0.0385[/tex]
The nearest answer for this case would be 3.1%
Step-by-step explanation:
Previous concepts
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
Solution to the problem
Let X the random variable of interest, on this case we now that:
[tex]X \sim Binom(n=20, p=0.71)[/tex]
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
For this case we can begin finding the probability P(X>10). If we find the individual probabilities we got:
[tex]P(X=11)=(20C11)(0.71)^{11} (1-0.71)^{20-11}=0.0563[/tex]
[tex]P(X=12)=(20C12)(0.71)^{12} (1-0.71)^{20-12}=0.1034[/tex]
[tex]P(X=13)=(20C13)(0.71)^{13} (1-0.71)^{20-13}=0.1558[/tex]
[tex]P(X=14)=(20C14)(0.71)^{14} (1-0.71)^{20-14}=0.1907[/tex]
[tex]P(X=15)=(20C15)(0.71)^{15} (1-0.71)^{20-15}=0.1867[/tex]
[tex]P(X=16)=(20C16)(0.71)^{16} (1-0.71)^{20-16}=0.1429[/tex]
[tex]P(X=17)=(20C17)(0.71)^{17} (1-0.71)^{20-17}=0.082[/tex]
[tex]P(X=18)=(20C18)(0.71)^{18} (1-0.71)^{20-18}=0.036[/tex]
[tex]P(X=19)=(20C19)(0.71)^{19} (1-0.71)^{20-19}=0.0087[/tex]
[tex]P(X=20)=(20C20)(0.71)^{20} (1-0.71)^{20-20}=0.00106[/tex]
And if we add the values we got:
[tex] P(X>10)= P(X=11) +.... +P(X=20) = 0.961525[/tex]
And if we use the complement rule the probability that "no more than 10 wear glasses for driving" we can do this:
[tex] P(X\leq 10) = 1- 0.961525= 0.0385[/tex]
The nearest answer for this case would be 3.1%
Money Flow The rate of a continuous money flow starts at $1000 and increases exponentially at 5% per year for 4 years. Find the present value and accumulated amount if interest earned is 3.5% compounded continuously.
Answer:
Present value = $4,122.4
Accumulated amount = $4,742
Step-by-step explanation:
Data provided in the question:
Amount at the Start of money flow = $1,000
Increase in amount is exponentially at the rate of 5% per year
Time = 4 years
Interest rate = 3.5% compounded continuously
Now,
Accumulated Value of the money flow = [tex]1000e^{0.05t}[/tex]
The present value of the money flow = [tex]\int\limits^4_0 {1000e^{0.05t}(e^{-0.035t})} \, dt[/tex]
= [tex]1000\int\limits^4_0 {e^{0.015t}} \, dt[/tex]
= [tex]1000\left [\frac{e^{0.015t}}{0.015} \right ]_0^4[/tex]
= [tex]1000\times\left [\frac{e^{0.015(4)}}{0.015} -\frac{e^{0.015(0)}}{0.015} \right][/tex]
= 1000 × [70.7891 - 66.6667]
= $4,122.4
Accumulated interest = [tex]e^{rt}\int\limits^4_0 {1000e^{0.05t}(e^{-0.035t}} \, dt[/tex]
= [tex]e^{0.035\times4}\times4,122.4[/tex]
= $4,742
The present value and interest accumulated would be as follows:
Present Value = $ 4,122.4
Interest Accumulated = $ 4742
Given that,
Principal at the beginning of money flow = $1,000
Exponential interest rate = 5% per year
Time Period = 4 years
So,
The accumulated money flow's worth = [tex]1000e^{0.05t}[/tex]
The current value of the money can be determined by [tex]\int\limits^4_0 1000e^{0.05t}(e^{-0.035t}) {} \, dt[/tex]
On solving, we get
The present value = $ 4,122.4
Interest Accumulated = $4,742
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A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions: a. In how many ways can one component of each type be selected
Answer:
120 ways
Step-by-step explanation: If there are three different categories of components, that is, component A, comprising four types; component B comprising, five types and component C comprising, six types. Three types are to be selected, that is, one type from each category. The number of ways one component of each type be selected is:
Component A * Component B * Component C = 4 X 5 X 6 = 120
Let R be the region bounded by y=x^2, x=1, and y=0. use the shell method to find the volume of the solid generated when R is revolved about the line x= 2
Answer:
[tex]V = \frac{5\pi}{6}[/tex] or 2.62
Step-by-step explanation:
Since our region (on the left) is bounded by x = 1 and x = 0 (where [tex]y = x^2 = 0[/tex], if we take center at x = 2 then our radius will range from 1 to 2 (x=1 to x = 0). We can use the following integration to calculate the volume using shell method
[tex]V = \int\limits^2_1 {2\pi r h} \, dr[/tex]
where r = 2 - x so x = 2 - r and [tex]h = y = x^2 = (2-r)^2[/tex] for [tex]1 \leq r \leq 2[/tex]
[tex]V = \int\limits^2_1 {2\pi r(2-r)^2} \, dr[/tex]
[tex]V = \int\limits^2_1 {2\pi r(4 - 4r + r^2)} \, dr[/tex]
[tex]V = 2\pi \int\limits^2_1 {r^3 - 4r^2 +4r} \, dr[/tex]
[tex]V = 2\pi\left[\frac{r^4}{4} - \frac{4r^3}{3} + 2r^2\right]^2_1[/tex]
[tex]V = 2\pi\left[\left(\frac{2^4}{4} - \frac{4*2^3}{3} + 2*2^2\right) - \left(\frac{1^4}{4} - \frac{4*1^3}{3} + 2*1^2\right)\right][/tex]
[tex]V = 2\pi(4 - 32/3 +8 - 1/4 + 4/3 - 2)[/tex]
[tex]V = \pi(20 - 56/3 - 1/2)[/tex]
[tex]V = \pi\frac{120 - 112 - 3}{6}[/tex]
[tex]V = \frac{5\pi}{6}[/tex] or 2.62
The branch of statistical studies called inferential statistics refers to drawing conclusions about sample data by analyzing the corresponding population. True or False
Answer: False
Step-by-step explanation:
In Inferential statistics the sample data is analysed rather than analysing the whole population, analysing the whole population may sometimes be impossible, like analysing the population of a whole country. Therefore, for inferential statistics, conclusions about the whole population are drawn by analyzing the corresponding sample data. The sample data are selected from the population and then analysed, the results can then be used to conclude on the whole population.
Determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0. (Enter your answers as a comma-separated list.) t2y'' − 2ty' + 2y = 0
Answer:
The only solution is r=2 or r=1
Step-by-step explanation:
Check the attachment
To determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0, substitute y = tr into the given differential equation and solve for r. The values of r that satisfy the equation are r = 1.
Explanation:To determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0, we need to substitute y = tr into the given differential equation and solve for r.
First, we find the first and second derivatives of y = tr:
y' = r
y'' = 0
Substituting these derivatives into the differential equation, we get:
t2(0) - 2t(r) + 2(tr) = 0
Simplifying the equation, we have:
(2 - 2r)t = 0
This equation is satisfied when t = 0 or r = 1. Therefore, the values of r for which the given differential equation has solutions of the form y = tr for t > 0 are r = 1.
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How expensive is Maui? A newspaper gave the following costs in dollars per day for a random sample of condominiums located throughout the island of Maui. 88 50 66 60 360 55 500 71 41 350 60 50 250 45 45 125 235 65 60 110 (a) Compute the mean, median, and mode for the data. (Round your answers to two decimal places.)
The mean (average) of the data set is approximately 110.40, the median (the middle value when the data set is arranged in order from least to greatest) is 65, and there is no mode (the most frequently occurring number) in the data set.
Explanation:The subject of your question is Mathematics, specifically in the field of Statistics. To compute the mean, median, and mode, you would do the following:
Add up all numbers in the data set and divide by the number of items in that set. This is the Mean. Arrange the data set from smallest to largest and find the middle value. If there is an even number of items in the data set, the median is the average of the middle two numbers. This is the Median. The number that appears most frequently in your data set is the Mode.For the given data set your mean is approximately 110.40 , the median is 65 and there is no mode as no numbers in the data set are repeated.
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You are applying for a job at two companies. Company A offers starting salaries with mu equals μ=$27,000 and σ=$3,000. Company B offers starting salaries with μ=$27,000 and sσ=$7,000. From which company are you more likely to get an offer of 33,000 ormore?
Answer:
For the company A : [tex] P(X>33000)= P(Z> \frac{33000-27000}{3000}) =P(Z>2) = 1-P(Z<2)= 0.0228[/tex]
For the company B: [tex] P(X>33000)= P(Z> \frac{33000-27000}{7000}) =P(Z>0.857) = 1-P(Z<0.857)= 0.196[/tex]
So as we can see we have a higher probability for the company B.
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".
Company A
Let X the random variable that represent the salaries of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(27000,3000)[/tex]
Where [tex]\mu=27000[/tex] and [tex]\sigma=3000[/tex]
For this case if we find the probability for [tex] P(X>33000)[/tex] using the z score given by:
[tex] z= \frac{x -\mu}{\sigma}[/tex]
And if we use this formula we got:
[tex] P(X>33000)= P(Z> \frac{33000-27000}{3000}) =P(Z>2) = 1-P(Z<2)= 0.0228[/tex]
Company B
Let X the random variable that represent the salaries of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(27000,7000)[/tex]
Where [tex]\mu=27000[/tex] and [tex]\sigma=7000[/tex]
For this case if we find the probability for [tex] P(X>33000)[/tex] using the z score given by:
[tex] z= \frac{x -\mu}{\sigma}[/tex]
And if we use this formula we got:
[tex] P(X>33000)= P(Z> \frac{33000-27000}{7000}) =P(Z>0.857) = 1-P(Z<0.857)= 0.196[/tex]
The company from which we are more likely to get an offer of $33,000 or more is determined by comparing the probabilities of receiving such an offer from each of the two companies. This probability is determined using a concept called the z-score, with which we can convert any data point into a common scale, which makes comparison easier.
1. Compute the z-scores for obtaining this salary for both companies. The z-score formula is (X - μ) / σ where X is the data point we are interested in, μ stands for the mean amount of the starting salary, and σ is the standard deviation.
For Company A:
z = (33000 - 27000) / 3000
We deduct the average salary for Company A (μ = 27000) from the target salary (X = 33000) and divide it by the standard deviation (σ = 3000).
For Company B:
Similarly, the z score is calculated by deducting the average salary for Company B (μ = 27000) the from target salary (X = 33000) and dividing by the standard deviation (σ = 7000).
2. Now we have the z-scores for both companies. The next step is to use the z-score to calculate the probability of getting an offer of $33,000 or more from both companies. We can do this by using the cumulative distribution function (CDF) which gives us the probability that a random variable is less than or equal to a certain value. But since we want the probability that the salary is more than $33,000, we subtract the result from one (1 - CDF(z)).
3. We compare these probabilities. The company with the higher probability value will be the one from which we are more likely to get an offer of $33,000 or more.
4. Based on these calculations, you are more likely to get an offer of $33,000 or more from Company B.
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Consider the demand curve Qd = 150 - 2P and the supply curve Qs = 50 + 3P. What is total expenditure at equilibrium? Make sure to round your answers to the nearest 100th percentage point. Also, do not write any symbol. For example, write 123.34 for $123.34. Let me know if you have questions. Good luck.
Answer:
20
Step-by-step explanation:
At equilibrium quantity demanded Qd equals quantity supply Qs
Qd = Qs (At equilibrium)
⇒ 150 - 2P = 50 + 3P
solving for P in the equation,
3P + 2P = 150 - 50
5P = 100
[tex]P = \frac{100}{5}[/tex]
P = 20
Positive-sequence components consist of three phasors with _____ magnitudes and _____ phase displacement in positive sequence; negativesequence components consist of three phasors with _____ magnitudes and _____ phase displacement in negative sequence; and zero-sequence components consist of three phasors with _____ magnitudes and _____ phase displacement.
Answer: The answers in order are: Equal, 120°, Equal, 120°, Equal, no
Step-by-step explanation:
The positive sequence components have equal magnitudes and 120° phase displacements in positive sequence. Their phase sequence is same as that of the system one. Also their phase rotations is same like the system.
The negative sequence components have equal magnitudes with same 120° phase displacements. Their phase sequence is opposite to that of system but their phase rotation is same like the system.
Zero sequence components have equal magnitude with no phase displacement. They behave like negative sequence in terms of phase sequence and phase rotation.
Answer:
Equal, 120°, Equal, 120°, Equal, no
Step-by-step explanation:
A construction company needs to remove 2 1/6 tons of dirt from a construction site. They can remove 710 tons of dirt each hour. What is the total number of hours it will take to remove the dirt?
Answer:
13/4260 tons
Step-by-step explanation:
We have the rate at which they remove tons of dirt per hour. We also know that total that needs to be removed. We can determine the time by dividing the amount of tons that need to be removed by the rate:
[tex]=(13/6)/\cdot{710}=13/4260[/tex]
will take 13/4260 hours to remove the dirt
Name the quadrant in which the angle below is located.
sine = 4 and cose is positive.
Answer:
first quadrant
Step-by-step explanation:
To know this we have to take into account that for the breast we have to look at the y-axis and in the cosine we will look at the x-axis
We have the positive and the negative part of the x-axis, now we notice that it says that the cosine is positive, then the quadrant will be with the positive x, these quadrants can be the first or the fourth.
Now if we look at sine = 4 is positive, this means that the axis y will also be positive, this means that it will be between the first and second quadrant.
Now if we look to meet the conditions of the sine and cosine, the angle has to be found in the first quadrant
Donna de paul is raising money for the homeless. She discovers that each church group requires 2 hours of let her writing and 1 hour of follow-up while for each labor union she needs 2 hours of letter written and 3 hours of follow-up. Donna can raise $150 from each church group and $200 from each Union Local and she has a maximum of 20 hours of letters written and a maximum of 16 hours of follow-up available per month. Determine the most profitable mixture of group she should contact in the most money she can lose in a month.
z=()x1+()x2
Answer:
x = 7
y = 3
z (max) = 4950/3 = 1650
Step-by-step explanation:
Let call
x numbers of church goup and
y numbers of Union Local
Then
First contraint
2*x + 2*y ≤ 20
Second one
1*x + 3*y ≤ 16
Objective Function
z = 150*x + 200*y
Then the system is
z = 150*x + 200*y To maximize
Subject to:
2*x + 2*y ≤ 20
1*x + 3*y ≤ 16
x ≥ 0 y ≥ 0
We will solve by using the Simplex method
z - 150 *x - 200*y = 0
2*x + 2*y + s₁ = 20
1*x + 3*y + 0s₁ + s₂ = 16
First Table
z x y s₁ s₂ Cte
1 -150 -200 0 0 = 0
0 2 2 1 0 = 20
0 1 3 0 1 = 16
First iteration:
Column pivot ( y column ) row pivot (third row) pivot 3
Second table
z x y s₁ s₂ Cte
1 -250/3 0 0 200/3 = 3200/3
0 - 4/3 0 -1 2/3 = -20/3
0 1/3 1 0 1/3 = -20/3
Second iteration:
Column pivot ( x column ) row pivot (second row) pivot -4/3
Third table
z x y s₁ s₂ Cte
1 0 0 750/12 700/6 = 4950/3
0 1 0 3/4 -1/2 = 7
0 0 1 -1/4 1/2 = 9/3
Donna should contact four church groups and four labor unions to maximize her fundraising, which would yield $1400. The maximum money she could 'lose' (or not earn) is $0 if she does not contact any group.
Explanation:We need to set up linear inequalities to represent the constraints described by Donna's situation. We will let x represent the number of church groups and y represent the number of labor unions she contacts.
The time required for letter writing can be described by the inequality: 2x + 2y <= 20, and the time for follow-ups is x + 3y <= 16.
We want to maximize the sum z = 150x + 200y which represents the total money she can raise.
To find the maximum, one method is to graph the feasible region defined by the constraints and find the vertices. Then evaluate the objective function at the vertices. In this case, the vertices are (0,0), (0,5), (8,0) and (4,4).
Upon calculation, contacting four church groups and four labor unions yields the highest amount of $1400.
As for the loss, given the nature of the task, the most money she could 'lose' in a month is just not raising any money at all, which would be $0 if she doesn't contact any group.
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Vectors are quantities that possess both magnitude and direction. In engineering problems, it is best to think of vectors as arrows, and usually it is best to manipulate vectors using components. In this tutorial, we consider the addition of two vectors using both of these techniques. Consider two vectors AAA_evec and BBB_evec that have lengths AAA and BBB, respectively. Vector BBB_evec make
Answer:
Check below
Step-by-step explanation:
Vectors are quantities that possess both magnitude and direction. In engineering problems, it is best to think of vectors as arrows, and usually it is best to manipulate vectors using components. In this tutorial, we consider the addition of two vectors using both of these techniques. Consider two vectors [tex]\vec{A}[/tex]and [tex]\vec{B}[/tex] that have lengths A and B, respectively. Vector [tex]\vec{B}[/tex] makes an angle?
1) Vector [tex]\vec{B}[/tex] makes an angle?
Yes, it does. Vector [tex]\vec{B}[/tex] makes an angle with [tex]\vec{A}[/tex], since both have the same origin and different direction.
2) From the direction of A.(Figure 1)In vector notation, the sum is represented by [tex]\vec{C}=\vec{A}+\vec{B}[/tex] where [tex]\vec{C}[/tex] is a new vector that is the sum of [tex]\vec{A}[/tex] and [tex]\vec{B}[/tex]. Find C, the length of C, which is the sum of A and B.
C is the resultant vector of this sum of vectors([tex](\vec{C}=\vec{A}+\vec{B})[/tex]
The length of c is found through the law of cosines, after projecting, vector a.
(Check 3rd picture)
2.2) The other technique to add vectors is to write them. As C is the resultant vector then we have
[tex]\vec{A}=\left \langle a_{1},a_{2} \right \rangle \vec{B}=\left \langle b_{1},b_{2} \right \rangle \vec{C}=\left \langle a_{1}+b_{1}, a_{2}+b_{2}\right \rangle[/tex]
In physics, vectors are quantities possessing both magnitude and direction. They can be represented as arrows on a graph, with their length and direction corresponding to the vector's magnitude and direction. Vectors are added geometrically using the head-to-tail method or analytically, where they are broken down into components, and these components are added separately.
Explanation:This can be visualized as an arrow, where its length corresponds to the magnitude, and its direction is represented by the way the arrow points. Some examples of vectors include displacement, velocity, and force.
When adding vectors, there are two methods to consider: geometric or component wise method. The geometric method involves representing the vectors as arrows on a graph and adding them using the head-to-tail method. The component-wise method involves breaking down the vector into its x and y components, and adding these components separately.
For instance, if you have two vectors A and B, vector A can be broken down into its x and y components: A_x and A_y. Likewise, vector B can be broken down into B_x and B_y. Simply add corresponding components to yield the resultant vector, R_x = (A_x + B_x) and R_y = (A_y + B_y)
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The Dow Jones Industrial Average has had a mean gain of 432 pear year with a standard deviation of 722. A random sample of 40 years is selected. What is the probability that the mean gain for the sample was between 250 and 500?
Answer:
66.98% probability that the mean gain for the sample was between 250 and 500.
Step-by-step explanation:
To solve this problem, it is important to know the Normal probability distribution and the Central limit theorem.
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].
In this problem, we have that:
[tex]\mu = 432, \sigma = 722, n = 40, s = \frac{722}{\sqrt{40}} = 114.16[/tex]
What is the probability that the mean gain for the sample was between 250 and 500?
This is the pvalue of Z when X = 500 subtracted by the pvalue of Z when X = 250.
So
X = 500
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{500 - 432}{114.16}[/tex]
[tex]Z = 0.6[/tex]
[tex]Z = 0.6[/tex] has a pvalue of 0.7257.
X = 250
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{250 - 432}{114.16}[/tex]
[tex]Z = -1.59[/tex]
[tex]Z = -1.59[/tex] has a pvalue of 0.0559.
So there is a 0.7257 - 0.0559 = 0.6698 = 66.98% probability that the mean gain for the sample was between 250 and 500.
The probability that the mean gain for a sample of 40 years is between 250 and 500 is approximately 0.6698, or 66.98%.
To determine the probability that the mean gain for a random sample of 40 years of the Dow Jones Industrial Average is between 250 and 500, we will use the concept of the sampling distribution of the sample mean. The steps involved are:
State the given information:
- Population mean [tex](\(\mu\))[/tex] = 432
- Population standard deviation [tex](\(\sigma\))[/tex] = 722
- Sample size n = 40
Find the standard error of the mean (SEM):
The standard error of the mean is calculated as:
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{722}{\sqrt{40}} \approx 114.2 \][/tex]
Convert the sample means to z-scores:
To find the probability that the sample mean [tex](\(\bar{x}\))[/tex] is between 250 and 500, we convert these values to z-scores using the formula:
[tex]\[ z = \frac{\bar{x} - \mu}{\text{SEM}} \] For \(\bar{x} = 250\): \[ z = \frac{250 - 432}{114.2} \approx \frac{-182}{114.2} \approx -1.59 \] For \(\bar{x} = 500\):[/tex]
[tex]\[ z = \frac{500 - 432}{114.2} \approx \frac{68}{114.2} \approx 0.60 \][/tex]
Find the probability corresponding to the z-scores:
Using the standard normal distribution table or a calculator, we find the probabilities corresponding to these z-scores.
[tex]\(z = -1.59\), \(P(Z \leq -1.59) \approx 0.0559\) \\\(z = 0.60\), \(P(Z \leq 0.60) \approx 0.7257\)[/tex]
Calculate the probability that the mean gain is between 250 and 500:
\[tex][ P(250 \leq \bar{x} \leq 500) = P(Z \leq 0.60) - P(Z \leq -1.59) \] \[[/tex]
= 0.7257 - 0.0559 = 0.6698
Conclusion:
The probability that the mean gain for a sample of 40 years is between 250 and 500 is approximately 0.6698, or 66.98%.
A baseball enthusiast believes pitchers who strike out a lot of batters also walk a lot of batters. He reached this conclusion by going to the library and examining the records of all major league pitchers between 1990 and 1995. What type of study is his decision based on? A) B) C) An observational study based on a sample survey D) An experiment. Anecdotal evidence. An observational study based on available data.
Answer:
D) An observational study based on available data.
Step-by-step explanation:
This is an observational study based on available data.
If it had been on a sample, he would take a sample of a few pitchers, and not studied the statistics of all major league pitchers during those seasons.
It is not an anecdotal evidence, because an anecdotal evidence is something without study, just an impression.
It is not an experiment, because he just studies(observes, that is why it is an observational study) the data, he does not change anything about the pitchers.
So the correct answer is:
D) An observational study based on available data.
The baseball enthusiast's decision is based on an observational study based on available data.
Explanation:The baseball enthusiast's decision is based on A) an observational study based on available data. In this case, the enthusiast examined the records of all major league pitchers between 1990 and 1995. This observational study involved collecting and analyzing data that was already available, without manipulating any variables or conducting an experiment.
A bottling plant fills one-gallon jugs with milk. The label on a one gallon jug of milk states that the volume of milk is 128 fluid ounces (fl.oz.) Federal law mandates that the jug must contain no less than the stated volume. The actual amount of milk in the jugs is normally distributed with mean µ = 129 fl. Oz. and standard deviation ơ = 0.8 fl. Oz. Use this information to answer below questions.
1. Find the z-score corresponding to a jug containing 128 fl. Oz. of milk?
2. What is the probability that a randomly selected jug will contain less than 128 fl. Oz. of milk?
Final answer:
The z-score for a jug with 128 fl. Oz. of milk and the probability of a jug containing less than 128 fl. Oz. are calculated using the mean and standard deviation provided.
Explanation:
1. Find the z-score corresponding to a jug containing 128 fl. Oz. of milk:
Calculate the z-score using the formula: z = (x - µ) / ơSubstitute the values: z = (128 - 129) / 0.8Calculate to find the z-score.2. Probability that a jug contains less than 128 fl. Oz. of milk:
Convert the value to a z-score using the formula.Refer to the Z-table or calculator to find the probability.A house hunter on Long Island estimates that 20% of the available houses in her price range are in acceptable condition. Furthermore, she has time to look at only one house each week. What is the probability that she will find an acceptable house in the first two weeks that she looks (round off to second decimal place)?
Answer:
36% probability that she will find an acceptable house in the first two weeks that she looks.
Step-by-step explanation:
For each house, there is a 20% probability of it being acceptable. And a 100-20 = 80% of not being acceptable
What is the probability that she will find an acceptable house in the first two weeks that she looks (round off to second decimal place)?
She only looks one house a week.
So this is the same as the probability of taking two or less weeks to find a house.
The are two outcomes
Finding an acceptable house in the first week, with 20% probability
Not finding an acceptable house in the first week, with 80% probability, and then finding an acceptable house in the second week, with 20% probability.
Probability:
[tex]P = 0.2 + 0.8*0.2 = 0.36[/tex]
36% probability that she will find an acceptable house in the first two weeks that she looks.
The correct answer is 0.36 or 36%.
To solve this problem, we need to calculate the probability that the house hunter does not find an acceptable house in the first two weeks and then subtract this probability from 1 to find the probability that she does find an acceptable house within that time frame.
Let's denote the probability of finding an acceptable house in a given week as[tex]\( p \)[/tex] and the probability of not finding an acceptable house in a given week as [tex]\( q \)[/tex]. Since 20% of the houses are in acceptable condition, [tex]\( p = 0.20 \) and \( q = 1 - p = 1 - 0.20 = 0.80 \).[/tex]
The probability that the house hunter does not find an acceptable house in the first week is[tex]\( q \).[/tex] The probability that she does not find an acceptable house in the first two weeks is the product of the probabilities that she does not find one in each of the two weeks separately, which is [tex]\( q \times q \) or \( q^2 \).[/tex]
Now, we calculate [tex]\( q^2 \):[/tex]
[tex]\[ q^2 = (0.80)^2 = 0.64 \][/tex]
This is the probability that she will not find an acceptable house in the first two weeks. To find the probability that she will find an acceptable house in the first two weeks, we subtract this value from 1:
[tex]\[ \text{Probability of finding an acceptable house in two weeks} = 1 - q^2 \][/tex]
[tex]\[ \text{Probability of finding an acceptable house in two weeks} = 1 - 0.64 \][/tex]
[tex]\[ \text{Probability of finding an acceptable house in two weeks} = 0.36 \][/tex]
A Gift for You makes floral arrangements and fruit baskets. The small business has a maximum of 40 hours per week available in the assembly department and a maximum of 10 hours per week in the packaging department. Each floral arrangement takes 20 minutes to assemble and 6 minutes to package. Each fruit basket takes 15 minutes to assemble and 2 minutes to package. The profit for each floral arrangement is $50 and the profit for each fruit basket is $35. The company wants to maximize their profit.a. Set up the linear programming problem by writing the objective function as well as the system of constraints. b. How many floral arrangements and fruit baskets must be sold to maximize profit? c. What is the maximum profit? Let x the number of floral arrangements and y the number of fruit baskets.Give the objective function a. Max P= 50x+ 35y b. Max P= 20x + 6y c. Min P= 15x+2yd. Max P= 35x +20ye. Min P= 35x+20yGive a constrainta. 20x+ 6y <=40b. 20x+15y<=40c. 6x+2y<=10d. 20x+ 15y<=24,00e. 6x+2y>=600How many of each should they sell to maximize their profit?a. 84 Floral Arrangements and 48 Fruit Baskets b. 0 Floral Arrangements and 160 Fruit Baskets c. 0 Floral Arrangements and 300 Fruit Baskets d. 100 Floral Arrangements and O Fruit Baskets e. 48 Floral Arrangements and 84 Fruit Baskets
Answer:
z(max) = 5000
x = 100
y = 0
Step-by-step explanation:
Let call
x floral arrangements and
y fruit baskets
Then Objective function is according to profits in each gift
z = 50*x + 35*y
Constraints:
1.- Hours available in Assembly department 40 in minutes is 2400 minutes
20*x + 15*y ≤ 2400
2.- Hours available in packaging department 10 in minutes is 600
6*x + 2*y ≤ 600
3.- x and y must be x ≥ 0 y ≥ 0
Then the system is:
z - 50*x - 50*y = 0 To maximize subject to:
20*x + 15*y ≤ 2400
6*x + 2*y ≤ 600
x ≥ 0 y ≥ 0
Simplex Method:
z x y s₁ s₂ Cte
1 -50 -35 0 0 = 0
0 20 15 1 0 = 2400
0 6 2 0 1 = 600
First iteration: 6 is a pivot we dvede R3 by 6
z x y s₁ s₂ Cte
1 0 15 0 50/6 = 5000
0 0 -50/6 -1 20/6 = -400
0 20 15 1 0 = 2400
0 1 2/6 0 1/6 = 100
We have done no negative number in the objective function we stop iteration and
z(max) = 5000
x = 100
y = 0
to add to R1 50*R3 [ 0 50 100/6 0 50/6 5000
to add to R2 20*R3 [ 0 20 40/6 0 20/6 2000
Preston and Joel are both solving the equation 2x=14. Preston is not sure what to do because he does not know a power of 2 that equals 14. Joel uses his calculator to graph y=2x and y=14 and find the point of intersection. Will Joel's method work?
Answer:
yes
Step-by-step explanation:
You can always separate an equation into two parts and see where those graphs intersect.
Joel's method works well.
_____
Additional comments
Preston should know that the invention of logarithms makes it easy to solve equations like this. x = log₂(14) = log(14)/log(2) ≈ 3.8073549.
As for Joel's method, I prefer to subtract the right side to get the equation ...
2^x -14 = 0
Then graphing y = 2^x -14, I look for the x-intercept. Most graphing calculators make it easy to find x- and y-intercepts. Not all make it easy to find points of intersection between different curves.
Answer:
Yes, the graph intersects around (3.807,14), so 3.807 is a good estimate of the solution to 2^x=14.
Step-by-step explanation:
What is the present value of a $50,000 decreasing perpetuity beginning in one year if the discount rate is 7% and the payments decline by 4% annually?
Answer:
Present Value = $1666666.67
Step-by-step explanation:
Present Value of a Growing Perpuity is calculated using the following formula
PV =D/(r - g)
Where D = Dividend
r = Discount Rate
g = Growth rate
D = $50,000
r = 7%
r = 7/100
r = 0.07
g = 4%
g = 4/100
g = 0.04
PV = D/(r-g)
Becomes
PV = $50,000/(0.07-0.04)
PV = $50,000/0.03
PV = $1,666,666.67
So the Present Value of the perpuity is $1,666,666.67
Suppose you and your 4 friends (5 people) are dressing up as the 6 main characters of the first Avengers movie: Iron Man, Hulk, Thor, Black Widow, Captain America and Hawkeye. (each question is independent of the others.) How many ways can you do this if all 5 people dress up as a different character?
Answer:
720
Step-by-step explanation:
If every person has to choose a different character, the first person to choose a character has 6 options, the second has 5, the third has 4, the fourth has 3, and the last person has only two options. Therefore, the total number of ways you can do this if all 5 people dress up as a different character is:
[tex]n=6*5*4*3*2\\n=720[/tex]
There are 720 ways.
Final answer:
There are 720 different ways for 5 people to dress up as the 6 main Avengers characters; this combinatorics problem uses permutations to find the answer.
Explanation:
Combinations for Avengers Characters
The question is about calculations of combinations, which falls under the subject of Mathematics. More specifically, this is a combinatorics problem that can typically be found at a High School level. We want to find out how many different ways 5 people can dress up as any of the 6 main Avengers characters, assuming each person dresses up as a different character. To solve this, we can use the concept of permutations because the order in which we assign the characters to the 5 friends does matter.
In this case, we have 6 characters to choose from, and we want to assign these characters to 5 friends. We are therefore looking for the number of permutations of 6 characters taken 5 at a time, which is calculated using the formula:
P(n, k) = n! / (n - k)!
Here, 'n' is the total number of characters, and 'k' is the number of people to dress up. Therefore, we have:
P(6, 5) = 6! / (6 - 5)! = 6! / 1! = 720 / 1 = 720
There are 720 different ways for the 5 friends to dress up as the Avengers characters.
A basketball player shoots a basketball with an initial velocity of 15 ft/sec. The ball is released from an initial height of 6.5 feet.
The function ℎ()=−162+0+h0 models the height, in feet, of an object after t seconds. v0 is the initial velocity of the object, and h0 is the initial height of the object.
Part 1: Write a function that models the height of the basketball. Use your function to answer Parts 2-4.
Part 2: How long does it take for the basketball to hit the ground? Round your answer to the nearest hundredth. Show all of your work. You're welcome to use this quadratic formula calculator, but please explain your answer.
Part 3: When does the basketball reach its maximum height? Round your answer to the nearest hundredth. Show all of your work and explain your answer.
Part 4: What is the maximum height of the basketball? Round your answer to the nearest hundredth. Show all of your work and explain your answer.
Answer:
∞
Step-by-step explanation:
A baseball team plays in a stadium that holds 56000 spectators. With the ticket price at $11 the average attendence has been 21000. When the price dropped to $9, the average attendence rose to 28000. Assume that attendence is linearly related to ticket price.
(a) Find the demand function, assuming that it is linear.
(b) How should ticket prices be set to maximize revenue?
Answer:
(a) [tex]p(x) = 17-\frac{x}{3,500}[/tex]
(b) $8.50
Step-by-step explanation:
(a) The slope of the demand function, p(x), is determined by:
[tex]m=\frac{11-9}{21,000-28,000}=-\frac{1}{3,500 }[/tex]
Applying the point (21,000; 11) to the general linear equation formula gives us the demand function:
[tex]p(x) - 11 = -\frac{1}{3,500}*(x-21,000)\\p(x) = 17-\frac{x}{3,500}[/tex]
(b) The revenue function, r(x), is given by:
[tex]r(x) =x*p(x) = 17x-\frac{x^2}{3,500}[/tex]
The value of x for which the derivate of the revenue function is zero gives us the attendance for which revenue is maximized:
[tex]\frac{dr(x)}{dx} =0= 17-\frac{2x}{3,500}\\x=29,750[/tex]
At an attendance of 29,750, the price is:
[tex]p = 17-\frac{29,750}{3,500}\\p=\$8.50[/tex]
Tickets should be set at a price of $8.50.
The demand function for the baseball team can be found using the given data points. The revenue function can be derived using the demand function, and its maximum can be determined by finding the critical points. The ticket price should be set at $8.50 to maximize revenue for the baseball team.
Explanation:(a) To find the demand function, we can use the point-slope form of a linear equation. Let's denote the ticket price as p and the attendance as a. We have two data points: (11, 21000) and (9, 28000). We can use the formula: a - a1 = m(p - p1) where (p1, a1) represents one of the given data points and m represents the slope. Let's use the first data point:
21000 - 28000 = m(11 - 9)
-7000 = 2m
m = -3500
So, the demand function is: a(p) = -3500p + b. To find the constant term b, we can substitute one of the data points into the equation. Let's use the first data point (11, 21000):
21000 = -3500(11) + b
21000 = -38500 + b
b = 59500
Therefore, the demand function is: a(p) = -3500p + 59500.
(b) Revenue is the product of the ticket price and the attendance. Let's denote the revenue as R and the ticket price as p. The revenue function can be written as: R(p) = p * a(p). Substituting the demand function into the revenue function, we have: R(p) = p * (-3500p + 59500). To maximize revenue, we can find the critical points of the function by taking the derivative and setting it equal to zero.
R'(p) = -7000p + 59500 = 0
-7000p = -59500
p = 8.5
The critical point is p = 8.5. To confirm that this gives a maximum, we can take the second derivative and check its sign.
R''(p) = -7000
Since the second derivative is negative, the critical point p = 8.5 corresponds to the maximum revenue. Therefore, the ticket price should be set at $8.50 to maximize revenue.
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