A concert promoter produces two kinds of souvenir shirt, one kind sells for $18 ad the other for $25. The company determines, the total cost, in thousands of dollars, of producting x thousand of the $18 shirt and y thousand of the $25 shirt is given by C(x,y)=4x^2-6xy+3y^2+20x+19y-12. How many of each type of shirt must be produced and sold in order to maximize profit?

Question
Multivariable functions
asked 2021-01-05
A concert promoter produces two kinds of souvenir shirt, one kind sells for $18 ad the other for $25. The company determines, the total cost, in thousands of dollars, of producting x thousand of the $18 shirt and y thousand of the $25 shirt is given by
\(\displaystyle{C}{\left({x},{y}\right)}={4}{x}^{{2}}-{6}{x}{y}+{3}{y}^{{2}}+{20}{x}+{19}{y}-{12}.\)
How many of each type of shirt must be produced and sold in order to maximize profit?

Answers (1)

2021-01-06
Revenue function
\(\displaystyle{R}{\left({x},{y}\right)}={18}{x}+{25}{y}\)
Profit function
\(\displaystyle{C}{\left({x},{y}\right)}={R}{\left({x},{y}\right)}-{C}{\left({x},{y}\right)}\)
\(\displaystyle{C}{\left({x},{y}\right)}=\cdot{18}{x}+{25}{y}{)}-{\left({4}{x}^{{2}}-{6}{x}{y}+{3}{y}^{{2}}+{20}{x}+{19}{y}-{12}\right)}\)
\(\displaystyle{C}{\left({x},{y}\right)}=-{4}{x}^{{2}}+{6}\times{y}-{3}{y}^{{2}}-{2}{x}+{6}{y}+{12}\)
First we will find the critical point. Now,
\(\displaystyle{P}_{{x}}{\left({x},{y}\right)}=-{8}{x}+{6}{y}-{2}\)
\(\displaystyle{P}_{{y}}{\left({x},{y}\right)}={6}{x}-{6}{y}+{6}\)
\(\displaystyle{P}_{{\times}}{\left({x},{y}\right)}=-{8}\)
\(\displaystyle{P}_{{{y}{y}}}{\left({x},{y}\right)}=-{6}\)
\(\displaystyle{P}_{{y}}{\left({x},{y}\right)}={0}\)
6x-6y+6=0
y=x+1
And \(\displaystyle{P}_{{x}}{\left({x},{y}\right)}={0}\)
6y-8x-2=0
6(x+1)-8x=2
x=2 (i)
So, y=x+1
y=2 (ii)
Hence, the critical point is (2,3)
Since \(\displaystyle{f}_{{\times}},{f}_{{{y}{y}}},{\quad\text{and}\quad}{f}_{{{x}{y}}}\) all are constant so we will need to pkug critical point to function
\(\displaystyle{D}{\left({x},{y}\right)}={f}_{{\times}}{f}_{{{y}{y}}}-{{f}_{{{x}{y}}}^{{2}}}\)
\(\displaystyle={\left(-{8}\right)}{\left(-{6}\right)}-{\left(-{6}\right)}^{{2}}\)
=12>0
Since, D>0, and \(\displaystyle{f}_{{\times}}{<}{0}\)</span>
The critical point P(2,3) has a relative maximum.
And the maximum value of profit function
\(\displaystyle{P}{\left({2},{3}\right)}=-{4}{\left({2}\right)}^{{2}}+{6}\cdot{2}\cdot{3}-{3}{\left({3}\right)}^{{2}}-{2}\cdot{2}+{6}\cdot{3}+{12}\)
\(\displaystyle{P}{\left({2},{3}\right)}={19}\)
The maximum profit of $19000 will be earned if 2000 shirt of $18 and 3000 shirts of $25 are produced and sold.
0

Relevant Questions

asked 2020-12-30
A company produces two products A and B, which have profits of $9 and $7. Each unit of product must be processed on two assembly lines where the required production times are a follows:
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\left|{c}\right|}\right\rbrace}{h}{l}\in{e}\text{Product}&\text{Line 1}&\text{Line 2}\backslash{h}{l}\in{e}{A}&{12}&{4}\backslash{h}{l}\in{e}\text{Total hours}&{60}&{40}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
a. formulate a linear programming model to determine the optimal product mix that will maximize profit.
b. transform this model into standard form.
c. identify the amount of unused resources(i.e., slack)at each of the graphical extreme points.
d. what would be the effect on the optimal if the production time on line 1 were reduced to 40 hours.
c. What would be the effect on the optimal solution if the profit for product B were increased from $7 to $15? To$20?
asked 2021-03-05
Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold.
\(\begin{array}{|c|c|} \hline Number\ N & Price\ p\\ \hline 200 & 53.00\\ \hline 250 & 52.50\\\hline 300 & 52.00\\ \hline 350 & 51.50\\ \hline \end{array}\)
(a) Find a formula for p in terms of N modeling the data in the table.
\(\displaystyle{p}=\)
(b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month.
\(\displaystyle{R}=\)
Is R a linear function of N?
(c) On the basis of the tables in this exercise and using cost, \(\displaystyle{C}={35}{N}+{900}\), use a formula to express the monthly profit P, in dollars, of this manufacturer as a function of the number of widgets produced in a month.
\(\displaystyle{P}=\)
(d) Is P a linear function of N?
asked 2020-11-01
You decide to make and sell two different gift baskets at your local outdoor market. Basket A contains 3 cookies, 6 chocolates, and 2 jars of jam and makes a profit of $12. Basket B contains 6 cookies, 3 chocolates, and 2 jars of jam and makes a profit of $15. You have just made 48 cookies, 36 chocolates, and 18 jars of jam. How many of each type of gift basket should you make to maximize the profit? a) State what you assign to x and y. Write the objective function. b) Write the three constraint inequalities. c) Find the axes intercepts of each of the above inequalities.
asked 2020-11-08
For a certain product, the revenue is given by R = 40: and the cost is given by C = 20x + 1600. To obtain a profit, the revenue must be greater than the cost. For what values of x will there be a profit?
asked 2020-12-02
A manager estimates a band's profit p for a concert by using the function p(t)=-200t^2+2500t-c, where t is the price per ticket and c is the band's operation cost. The table shows the band's operating cost at three different concert locations. What range of ticket prices should the band charge at each location in order to make a profit of at least $1000 at each concert?
Band's Costs
Location Operating Cost
Freemont Park $900
Saltillo Plaza $1500
Riverside Walk $2500
asked 2021-02-01
A coffee merchant sells three blends of coffee. A bag of houseblend contains 300 grams of Colombian beans, 50 grams of Kenyanbeans, and 150 grams of French roast beans. A bag of specialblend contains 200 grams of Colombian beans, 200 grams of Kenyanbeans, and 100 grams of French roast beans. A bag of Gourmet blend contains 100 grams of Colombian beans, 350 grams of Kenyanbeans and 50 grams of French roast beans. Merchant has on hand 30kg of Colombian beans, 15 kg of Kenyan beans and 15 kg of French roast beans. One bag of the house blend produces profit of $0.50 , onebag of the Special blend produces a profit of $1.50, and one bag ofthe gourmet blend produces a profit of $2.00. How many bags of eachtype should the merchant prepare if he wants to use up all of thebeans and maximize his profit? What is the maximum profit?
asked 2020-10-23
The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P \((A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).\)
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
asked 2021-01-19
A company that makes thing-a-ma-bobs has a start up cost of $44055. It costs the company $1.97 to make each thing-a-ma-bob. The company charges $5.31 for each thing-a-ma-bob. Let x denote the number of thing-a-ma-bobs produced.
Write the cost function for this company. C(x) =
Write the revenue function for this company. R(x) =
What is the minumum number of thing-a-ma-bobs that the company must produce and sell to make a profit?
asked 2021-03-11
Let's say the widget maker has developed the following table that shows the highest dollar price p. widget where you can sell N widgets. Number N Price p \(200 53.00\)
\(250 52.50\)
\(300 52.00\)
\(35051.50\) (a) Find a formula for pin terms of N modeling the data in the table. (b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in month as a function of the number N of widgets produced in a month. \(R=\) Is Ra linear function of N? (c) On the basis of the tables in this exercise and using cost, \(C= 35N + 900\), use a formula to express the monthly profit P, in dollars, of this manufacturer asa function of the number of widgets produced in a month \(p=\) (d) Is Pa linear function of N2 e. Explain how you would find breakeven. What does breakeven represent?
asked 2021-03-02
The cost in dollars to produce x youth baseball caps is C(x) = 4.3x + 75. The revenue in dollars from sales of x caps is \(\displaystyle{R}{\left({x}\right)}={25}{x}\).
(a) Write and simplify a function P that gives profit in terms of x.
(b) Find the profit if 50 caps are produced and sold.
...