Question 1

Find x and y such that   subject to the constraint   attains a minimum value. ​

Accepted Answer

A)    
B) ​   
C) ​   
D) ​   
E) ​   
A)    
B) ​   
C) ​   
D) ​   
E) ​

Question 2

Suppose that the profit from the sale of Kandy and Kreams is given by   dollars, where x is the number of pounds of Kandy and y is the number of pounds of Kreams. How many pounds of Kandy and Kreams must be sold to maximize profit? What is the maximum profit? Round your pounds answers to the nearest whole number. Round your max profit answer to the nearest cent. ​

Accepted Answer

A) 40 pounds of Kandy, 200 pounds of Kreams, max profit of $661.00 
B) 6 pounds of Kandy, 15 pounds of Kreams, max profit of $477.45 
C) 12 pounds of Kandy, 30 pounds of Kreams, max profit of $865.80 
D) 20 pounds of Kandy, 100 pounds of Kreams, max profit of $1,620.00 
E) 60 pounds of Kandy, 300 pounds of Kreams, max profit of $315.34 
A) 40 pounds of Kandy, 200 pounds of Kreams, max profit of $661.00 
B) 6 pounds of Kandy, 15 pounds of Kreams, max profit of $477.45 
C) 12 pounds of Kandy, 30 pounds of Kreams, max profit of $865.80 
D) 20 pounds of Kandy, 100 pounds of Kreams, max profit of $1,620.00 
E) 60 pounds of Kandy, 300 pounds of Kreams, max profit of $315.34

Question 3

If   , find   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 4

The total cost (in dollars) of producing 1 unit of a product is given by   , where x represents the cost per pound of raw materials and y represents the hourly rate for labor. The present cost for raw materials is $15 per pound and the present hourly rate for labor is $11. Indicate how to use the cost function C to determine how an increase of $1 per pound for raw materials will affect the total cost. Round your answer to two decimal places. ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 5

If   is the utility function for goods X and Y, the marginal utility of X is   and the marginal utility of Y is   . If   , find the marginal utility of X and Y.

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 6

Find the maximum value of the function   subject to   ,   ,   . ​

Accepted Answer

A) ​96,768 
B) ​4,032 
C) ​169,344 
D) ​12,160 
E) ​24,192 
A) ​96,768 
B) ​4,032 
C) ​169,344 
D) ​12,160 
E) ​24,192

Question 7

A biologist has conjectured that the weight of a certain lizard in pounds as a function of age A in years and length L in inches is given by   . If a lizard were to remain 13 inches long as it aged from 4 to 5 years old, then determine how much the lizard's weight would be predicted to increase over that year. Round to three decimal places. ​

Accepted Answer

A) ​  
Lbs 
B) ​  
Lbs 
C) ​  
Lbs 
D)   
Lbs 
E) ​  
Lbs 
A) ​  
Lbs 
B) ​  
Lbs 
C) ​  
Lbs 
D)   
Lbs 
E) ​  
Lbs

Question 8

Suppose that the following table shows the average hourly earnings for full-time production workers in various industries for selected years. Find the linear regression equation for hourly earnings as a function of time (with   representing 1970). Round numerical values in your answer to four decimal places. ​   ​

Accepted Answer

A)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage) 
B)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage) 
C)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage) 
D)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage) 
E)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage) 
A)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage) 
B)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage) 
C)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage) 
D)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage) 
E)   
(where  
Is the number of years after 1970 and  
Is the average hourly wage)

Question 9

Suppose that the utility function for two goods X and Y is given by   , and a consumer purchases 2 units of X and 6 units of Y. If the consumer purchases 2 units of Y, how many units of X must be purchased to retain the same level of utility? ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 10

If   , find   . ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 11

The cost (in dollars) of manufacturing one item is given by   , where x is the cost of 1 hour of labor and y is the cost of 1 pound of material. If the hourly cost of labor is $5, and the material costs $2 per pound, what is the cost of manufacturing one of these items? ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

Suppose that the number of tons of an agricultural product produced is given by   , where x is the number of hours of labor and y is the number of acres of the crop. Find the number of tons produced when   and   . Round your answer to the nearest number of tons. ​

Accepted Answer

A)   
Tons 
B)   
Tons 
C) ​  
Tons 
D) ​  
Tons 
E) ​  
Tons 
A)   
Tons 
B)   
Tons 
C) ​  
Tons 
D) ​  
Tons 
E) ​  
Tons

Question 13

The material for the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. Use Lagrange multipliers to find the dimensions of the box of largest volume that can be made for a fixed cost of 200.00. Round your answers to two decimal places. (Maximize   subject to   .) ​

Accepted Answer

A) Bottom dimensions:  
; height:   
B) Bottom dimensions:  
; height:   
C) Bottom dimensions:  
; height:   
D) Bottom dimensions:  
; height:   
E) Bottom dimensions:  
; height:   
A) Bottom dimensions:  
; height:   
B) Bottom dimensions:  
; height:   
C) Bottom dimensions:  
; height:   
D) Bottom dimensions:  
; height:   
E) Bottom dimensions:  
; height:

Question 14

Find the values for each of the dimensions of an open-top box of length x, width y, and height   (in inches) such that the box requires the least amount of material to make. ​

Accepted Answer

A) length = 8 inches width = 8 inches
Height = 4 inches 
B) length = 4 inches width = 8 inches
Height = 8 inches 
C) length = 8 inches width = 4 inches
Height = 8 inches 
D) length = 4 inches width = 4 inches
Height = 16 inches 
E) length = 16 inches width = 4 inches
Height = 4 inches 
A) length = 8 inches width = 8 inches
Height = 4 inches 
B) length = 4 inches width = 8 inches
Height = 8 inches 
C) length = 8 inches width = 4 inches
Height = 8 inches 
D) length = 4 inches width = 4 inches
Height = 16 inches 
E) length = 16 inches width = 4 inches
Height = 4 inches

Question 15

Test for relative maximum and minimum. ​   ​

Accepted Answer

A) relative maximum at   
B) relative maximum at   
C) relative minimum at   
D) relative minimum at   
E) relative minimum at   
A) relative maximum at   
B) relative maximum at   
C) relative minimum at   
D) relative minimum at   
E) relative minimum at

Question 16

Suppose that an indifference curve for two goods, X and Y, has the equation   . If 30 units of X are purchased, how many units of Y must be purchased to remain on this indifference curve? ​

Accepted Answer

A)   
Units of Y. 
B) 57,600 units of Y. 
C) 64 units of Y. 
D) 87 units of Y. 
E) 101 units of Y. 
A)   
Units of Y. 
B) 57,600 units of Y. 
C) 64 units of Y. 
D) 87 units of Y. 
E) 101 units of Y.

Question 17

Suppose that the output Q (in units) of a certain company is   , where K is the capital expenditures in thousands of dollars and L is the number of labor hours. Find   and   when capital expenditures are $45,000 and the labor hours total 5,400. Round your answers to two decimal places. ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 18

If   , find   and   . ​

Accepted Answer

A)   
,   
B)   
,   
C)   
,   
D)   
,   
E)   
,   
A)   
,   
B)   
,   
C)   
,   
D)   
,   
E)   
,

Question 19

Suppose that the cost of producing x units at plant X is   dollars and that the cost of producing y units of the same product at plant Y is   dollars. If the firm that owns the plants has an order for 240 units, how many should it produce at each plant to fill this order and minimize its cost of production? ​

Accepted Answer

A) The production of 96 units at plant X and 144 units at plant Y minimizes the production costs. 
B) The production of 144 units at plant X and 96 units at plant Y minimizes the production costs. 
C) The production of 86 units at plant X and 154 units at plant Y minimizes the production costs. 
D) The production of 111 units at plant X and 129 units at plant Y minimizes the production costs. 
E) The production of 154 units at plant X and 86 units at plant Y minimizes the production costs. 
A) The production of 96 units at plant X and 144 units at plant Y minimizes the production costs. 
B) The production of 144 units at plant X and 96 units at plant Y minimizes the production costs. 
C) The production of 86 units at plant X and 154 units at plant Y minimizes the production costs. 
D) The production of 111 units at plant X and 129 units at plant Y minimizes the production costs. 
E) The production of 154 units at plant X and 86 units at plant Y minimizes the production costs.

Question 20

Find x and y such that   subject to the condition   attains a minimum value. ​

Accepted Answer

A)    
B) ​   
C) ​   
D) ​   
E) ​   
A)    
B) ​   
C) ​   
D) ​   
E) ​

Find x and y such that subject to the constraint attains a minimum value.

If , find .

If is the utility function for goods X and Y, the marginal utility of X is and the marginal utility of Y is . If , find the marginal utility of X and Y.

Find the maximum value of the function subject to , , .

Suppose that the utility function for two goods X and Y is given by , and a consumer purchases 2 units of X and 6 units of Y. If the consumer purchases 2 units of Y, how many units of X must be purchased to retain the same level of utility?

If , find .

The cost (in dollars) of manufacturing one item is given by , where x is the cost of 1 hour of labor and y is the cost of 1 pound of material. If the hourly cost of labor is $5, and the material costs $2 per pound, what is the cost of manufacturing one of these items?

Suppose that the number of tons of an agricultural product produced is given by , where x is the number of hours of labor and y is the number of acres of the crop. Find the number of tons produced when and . Round your answer to the nearest number of tons.

Find the values for each of the dimensions of an open-top box of length x, width y, and height (in inches) such that the box requires the least amount of material to make.

Test for relative maximum and minimum.

Suppose that an indifference curve for two goods, X and Y, has the equation . If 30 units of X are purchased, how many units of Y must be purchased to remain on this indifference curve?

Suppose that the output Q (in units) of a certain company is , where K is the capital expenditures in thousands of dollars and L is the number of labor hours. Find and when capital expenditures are $45,000 and the labor hours total 5,400. Round your answers to two decimal places.

If , find and .

Find x and y such that subject to the condition attains a minimum value.

Algebraic Concepts

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Filters

Exam 15: Functions of Two or More Variables

Find x and y such that subject to the constraint attains a minimum value. ​

If , find . ​

If is the utility function for goods X and Y, the marginal utility of X is and the marginal utility of Y is . If , find the marginal utility of X and Y.

Find the maximum value of the function subject to , , . ​

Suppose that the utility function for two goods X and Y is given by , and a consumer purchases 2 units of X and 6 units of Y. If the consumer purchases 2 units of Y, how many units of X must be purchased to retain the same level of utility? ​

If , find . ​

The cost (in dollars) of manufacturing one item is given by , where x is the cost of 1 hour of labor and y is the cost of 1 pound of material. If the hourly cost of labor is $5, and the material costs $2 per pound, what is the cost of manufacturing one of these items? ​

Suppose that the number of tons of an agricultural product produced is given by , where x is the number of hours of labor and y is the number of acres of the crop. Find the number of tons produced when and . Round your answer to the nearest number of tons. ​

Find the values for each of the dimensions of an open-top box of length x, width y, and height (in inches) such that the box requires the least amount of material to make. ​

Test for relative maximum and minimum. ​ ​

Suppose that an indifference curve for two goods, X and Y, has the equation . If 30 units of X are purchased, how many units of Y must be purchased to remain on this indifference curve? ​

Suppose that the output Q (in units) of a certain company is , where K is the capital expenditures in thousands of dollars and L is the number of labor hours. Find and when capital expenditures are $45,000 and the labor hours total 5,400. Round your answers to two decimal places. ​

If , find and . ​

Find x and y such that subject to the condition attains a minimum value. ​

Algebraic Concepts

Linear Equations and Functions

Quadratic and Other Special Functions

Matrices

Inequalities and Linear Programming

Exponential and Logarithmic Functions

Mathematics of Finance

Introduction to Probability

Further Topics in Probability; Data Description

Derivatives

Applications of Derivatives

Derivatives Continued

Indefinite Integrals

Definite Integrals: Techniques of Integration

Filters

Find x and y such that subject to the constraint attains a minimum value.

If , find .

Find the maximum value of the function subject to , , .

Suppose that the utility function for two goods X and Y is given by , and a consumer purchases 2 units of X and 6 units of Y. If the consumer purchases 2 units of Y, how many units of X must be purchased to retain the same level of utility?

If , find .

The cost (in dollars) of manufacturing one item is given by , where x is the cost of 1 hour of labor and y is the cost of 1 pound of material. If the hourly cost of labor is $5, and the material costs $2 per pound, what is the cost of manufacturing one of these items?

Suppose that the number of tons of an agricultural product produced is given by , where x is the number of hours of labor and y is the number of acres of the crop. Find the number of tons produced when and . Round your answer to the nearest number of tons.

Find the values for each of the dimensions of an open-top box of length x, width y, and height (in inches) such that the box requires the least amount of material to make.

Test for relative maximum and minimum.

Suppose that an indifference curve for two goods, X and Y, has the equation . If 30 units of X are purchased, how many units of Y must be purchased to remain on this indifference curve?

Suppose that the output Q (in units) of a certain company is , where K is the capital expenditures in thousands of dollars and L is the number of labor hours. Find and when capital expenditures are $45,000 and the labor hours total 5,400. Round your answers to two decimal places.

If , find and .

Find x and y such that subject to the condition attains a minimum value.