Question 1

Maximize $$p = 15 x + 14 y$$ subject to the following.   $$\begin{array} { l } 
2 x + 2 y \leq 8 \
2 x + 8 y \leq 16 \
x \geq 0 , y \geq 0
\end{array}$$

Accepted Answer

A)  $p = 60$ , $x = 4$ , $y = 4$ 
B)   $p = 60$ , $x = 0$ , $y = 4$ 
C)    $p = 60$ , $x = 4$ , $y = 0$ 
D)    $p = 30$ , $x = 4$ , $y = 0$ 
E)    $p = 30$ , $x = 0$ , $y = 4$ 
A)  $p = 60$ , $x = 4$ , $y = 4$ 
B)   $p = 60$ , $x = 0$ , $y = 4$ 
C)    $p = 60$ , $x = 4$ , $y = 0$ 
D)    $p = 30$ , $x = 4$ , $y = 0$ 
E)    $p = 30$ , $x = 0$ , $y = 4$

Question 2

Solve the linear programming problem. Maximize $P = 2 x + 8 y$ subject to $x + y \leq 4$ $2 x + y \leq 5$ $x \geq 0 , y \geq 0$

Accepted Answer

A)  $x = 0$ , $y = 5$ , $P = 40$ 
B)  $x = 1$ , $y = 3$ , $P = 26$ 
C)  $x = 1$ , $y = 0$ , $P = 26$ 
D)  $x = 2.5$ , $y = 0$ , $P = 5$ 
E)  $x = 0$ , $y = 4$ , $P = 32$ 
A)  $x = 0$ , $y = 5$ , $P = 40$ 
B)  $x = 1$ , $y = 3$ , $P = 26$ 
C)  $x = 1$ , $y = 0$ , $P = 26$ 
D)  $x = 2.5$ , $y = 0$ , $P = 5$ 
E)  $x = 0$ , $y = 4$ , $P = 32$

Question 3

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and no Vitamin C. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 45 percent of the U.S. Recommended Daily Allowance (RDA) of Vitamin C for infants. Each serving of Gerber Apple Banana Juice contains 60 calories and 120 percent of the RDA of Vitamin C for infants. The cereal costs 10 cents per serving, the dessert costs 53 cents per serving, and the juice costs 27 cents per serving. If you want to provide your child with at least 120 calories and at least 120 percent of the RDA of Vitamin C, how can you do so at the least cost  ?

Accepted Answer

A) 1 serving of cereal, 1 serving of juice, and no dessert 
B)  1 serving of dessert, 1 serving of juice, and no cereal 
C)  1 serving of cereal, 3 servings of dessert, and no juice 
D)  3 servings of dessert, 1 serving of juice, and no cereal 
E)  1 serving of cereal, 1 serving of dessert, and no juice 
A) 1 serving of cereal, 1 serving of juice, and no dessert 
B)  1 serving of dessert, 1 serving of juice, and no cereal 
C)  1 serving of cereal, 3 servings of dessert, and no juice 
D)  3 servings of dessert, 1 serving of juice, and no cereal 
E)  1 serving of cereal, 1 serving of dessert, and no juice

Question 4

Solve the standard minimization problem using duality.
​
Minimize $c = 4 s + t + 7 u$ ​
subject to $$\begin{aligned}
s + t + u & \geq 50 \
2 s + t & \geq 25 \
t + u & \geq 25 \
s \geq 0 , t \geq 0 , u & \geq 0
\end{aligned}$$ ​ $c =$ _________
​ $s =$ _________
​ $t =$ __________
​ $u =$ __________

Accepted Answer

The answer of Solve the standard minimization problem using duality.
​
Minimize...

Question 5

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and 11 grams of carbohydrates, and each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 21 grams of carbohydrates. You want to provide your child with at least 340 calories and at least 75 grams of carbohydrates. Draw the feasible region that shows the number of servings of cereal and number of servings of dessert that you can give your child. Find the corner points of the region.  
Let the number of servings of cereal be x, and the number of dessert be y.

Accepted Answer

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

Question 6

The political pollster Canter is preparing for a national election. It would like to poll at least 2,400 Democrats and 2,400 Republicans. Each mailing to the East Coast gets responses from 100 Democrats and 50 Republicans. Each mailing to the Midwest gets responses from 100 Democrats and 100 Republicans. And each mailing to the West Coast gets responses from 50 Democrats and 100 Republicans. Mailings to the East Coast cost $40 each to produce and mail, mailings to the Midwest cost $60 each, and mailings to the West Coast cost $50 each. How many mailings should Canter send to each area of the country to get the responses it needs at the least possible cost  What will it cost  ?

Accepted Answer

A) 16 mailings to the East Coast, none to the Midwest, 16 to the West Coast, the cost is $1,440 
B)  No mailings to the East Coast, 24 to the Midwest, none to the West Coast, the cost is $1,440 
C)  24 mailings to the East Coast, none to the Midwest, 24 to the West Coast, another solution resulting in the same cost is no mailings to the East Coast, 16 to the Midwest, none to the West Coast, the cost is $1,440 
D)  16 mailings to the East Coast, none to the Midwest, 16 to the West Coast, another solution resulting in the same cost is no mailings to the East Coast, 24 to the Midwest, none to the West Coast, the cost is $1,440 
E)  No mailings to the East Coast, 16 to the Midwest, none to the West Coast, the cost is $1,440 
A) 16 mailings to the East Coast, none to the Midwest, 16 to the West Coast, the cost is $1,440 
B)  No mailings to the East Coast, 24 to the Midwest, none to the West Coast, the cost is $1,440 
C)  24 mailings to the East Coast, none to the Midwest, 24 to the West Coast, another solution resulting in the same cost is no mailings to the East Coast, 16 to the Midwest, none to the West Coast, the cost is $1,440 
D)  16 mailings to the East Coast, none to the Midwest, 16 to the West Coast, another solution resulting in the same cost is no mailings to the East Coast, 24 to the Midwest, none to the West Coast, the cost is $1,440 
E)  No mailings to the East Coast, 16 to the Midwest, none to the West Coast, the cost is $1,440

Question 7

Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 720 grams of protein and at least 370 grams of fat. If chicken costs 10 cents per ounce and grain costs 1 cent per ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost  ?

Accepted Answer

A) 0 ounces of chicken and 360 ounces of grain 
B)  70 ounces of chicken and 0 ounces of grain 
C)  70 ounces of chicken and 10 ounces of grain 
D)  74 ounces of chicken and 0 ounces of grain 
E)  no solution 
A) 0 ounces of chicken and 360 ounces of grain 
B)  70 ounces of chicken and 0 ounces of grain 
C)  70 ounces of chicken and 10 ounces of grain 
D)  74 ounces of chicken and 0 ounces of grain 
E)  no solution

Question 8

You are thinking of making your mansion more energy efficient by replacing some of the light bulbs with compact fluorescent bulbs, and insulating part or all of your exterior walls. Each compact fluorescent light bulb costs $4 and saves you an average of $2 per year in energy costs, and each square foot of wall insulation costs $1 and saves you an average of $0.20 per year in energy costs. Your mansion has 200 light fittings and 2,000 sq. ft. of uninsulated exterior wall. To impress your friends, you would like to spend as much as possible, but save no more than $600 per year in energy costs (you are proud of your large utility bills). How many compact fluorescent light bulbs and how many square feet of insulation should you purchase  How much will you save in energy costs per year  ?

Accepted Answer

A) 100 fluorescent light bulbs, 2,000 square feet of insulation should be purchased, and $600 per year will be saved 
B)  200 fluorescent light bulbs, 1,000 square feet of insulation should be purchased, and $600 per year will be saved 
C)  200 fluorescent light bulbs, 0 square feet of insulation should be purchased, and $800 per year will be saved 
D)  0 fluorescent light bulbs, 2,000 square feet of insulation should be purchased, and $2,000 per year will be saved 
E)  no solution 
A) 100 fluorescent light bulbs, 2,000 square feet of insulation should be purchased, and $600 per year will be saved 
B)  200 fluorescent light bulbs, 1,000 square feet of insulation should be purchased, and $600 per year will be saved 
C)  200 fluorescent light bulbs, 0 square feet of insulation should be purchased, and $800 per year will be saved 
D)  0 fluorescent light bulbs, 2,000 square feet of insulation should be purchased, and $2,000 per year will be saved 
E)  no solution

Question 9

Your friend's portfolio manager has suggested two high-yielding stocks: Consolidated Edison (ED) and Royal Bank of Scotland (RBS-K). ED shares cost $40, yield 5.5% in dividends, and have a risk index of 1.0 per share. RBS-K shares cost $25, yield 7.5% in dividends, and have a risk index of 1.5 per share. You have up to $22,000 to invest and would like to earn at least $1,210 in dividends. How many shares (to the nearest whole number) of each stock should you purchase in order to meet your requirements and minimize the total risk index for your portfolio 
If no optimal solution exists enter no solution in each box.
?
Number of shares of ED = __________
?
Number of shares of RBS-K = __________

Accepted Answer

The answer of Your friend's portfolio manager has suggested two...

Question 10

Write down (without solving) the dual LP problem.
​
Minimize $c = s + 6 t + u$ subject to
​ $\begin{array} { l } 
2 s - t + v \geq 1,000 \
u - v \geq 2,000 \
s + t \geq 1,500 \
s \geq 0 , t \geq 0 , u \geq 0 , v \geq 0
\end{array}$ ​
Use the letters x, y, z, and w in the indicated order. Please, use the numbers in the expression for ​p in order they come in the given LP problem.
​
Maximize $p =$ __________ subject to the constraints.
​
Please enter the inequalities, separated with commas.

Accepted Answer

The answer of Write down (without solving) the dual LP...

Question 11

Graph the solution set to the inequality. Label coordinates of all corner points (if any).   $\left\{ \begin{array} { l } 
2 x + y \leq 16 \
x - 2 y \leq 8
\end{array} \right.$

Accepted Answer

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

Question 12

The Marketing Club at your college has decided to raise funds by selling three types of T-shirt: one with a single-color "ordinary" design, one with a two-color "fancy" design, and one with a three-color "very fancy" design. The club feels that it can sell up to 200 T-shirts. "Ordinary" T-shirts will cost the club $6 each, "fancy" T-shirts $8 each, and "very fancy" T-shirts $10 each, and the club has a total purchasing budget of $2,000. It will sell "ordinary" T-shirts at a profit of $4 each, "fancy" T-shirts at a profit of $5 each, and "very fancy" T-shirts at a profit of $4 each. How many of each kind of T-shirt should the club order to maximize profit 
?
The club should order __________ "ordinary" T-shirt(s), __________ "fancy" T-shirt(s) and __________ "very fancy" T-shirt(s).
?
What is the maximum profit the club can make 
?
The profit is $__________.

Accepted Answer

The answer of The Marketing Club at your college has...

Question 13

Use the simplex method to maximize $p = 2 x + y + 3 z + w + 2 v$ subject to the following.
​ $$\begin{array} { l } 
x + y \leq 3 \
y + z \leq 1 \
z + w \leq 3 \
w + v \leq 6
\end{array}$$ $$x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0 , v \geq 0$$ ​ $p =$ __________
​ $x =$ __________
​ $y =$ __________
​ $z =$ __________
​ $w ^ { \prime } =$ __________
​ $v =$ __________

Accepted Answer

13.2; 1.1;

Question 14

You are planning to invest up to $16,000 in IBM, HPQ, and AAPL shares. You desire to maximize your share of the companies' earnings but, for tax reasons, want to earn no more than $320 in dividends. Your broker suggests that because AAPL stocks pay no dividends, you should invest only in AAPL. Is she right  $$\begin{array} { | c | c | c | c | } 
\hline & \text { Price } & \begin{array} { c } 
\text { Dividend } \
\text { Yield }
\end{array} & \begin{array} { c } 
\text { Earnings } \
\text { per share }
\end{array} \
\hline \text { IBM } & \$ 80 & 1 \% & \$ 5 \
\hline \begin{array} { c } 
\text { HPQ (Hewlett } \
\text { Packard) }
\end{array} & \$ 20 & 2 \% & \$ 1 \
\hline \text { AAPL (Apple) } & \$ 70 & 0 \% & \$ 1 \
\hline
\end{array}$$

Accepted Answer

A) True 
 B)False

Question 15

Solve the standard minimization problem using duality.
​
Minimize $c = 0.3 s + 0.5 t$ ​
subject to $$\begin{array} { l } 
0.2 s + 0.1 t \geq 1 \
0.15 s + 0.3 t \geq 1.5 \
10 s + 10 t \geq 80 \
s \geq 0 , t \geq 0
\end{array}$$ ​ $c =$ __________
​ $s =$ __________
​ $t =$ __________

Accepted Answer

The answer of Solve the standard minimization problem using duality.
​
Minimize...

Question 16

Minimize  $c = 4 x + 7 y$  
Subject to
  $\begin{array} { l } 
8 x + 3 y \geq 30 \
2 x + 3 y \geq 12 \
x \geq 0 , y \geq 0
\end{array}$

Accepted Answer

A)  $c = 24$ , $x = 6$ , $y = 0$ 
B)  $c = 21$ , $x = 0$ , $y = 3$ 
C)  $c = 6$ , $x = 3$ , $y = 0$ 
D)  $c = 42$ , $x = 0$ , $y = 6$ 
E)  no solution 
A)  $c = 24$ , $x = 6$ , $y = 0$ 
B)  $c = 21$ , $x = 0$ , $y = 3$ 
C)  $c = 6$ , $x = 3$ , $y = 0$ 
D)  $c = 42$ , $x = 0$ , $y = 6$ 
E)  no solution

Question 17

Solve the standard minimization problem using duality.
​
Minimize $$c = 7 s + 6 t$$ ​
subject to $$\begin{array} { l } 
s + t \geq 12 \
7 s + t \geq 42 \
s \geq 0 , t \geq 0
\end{array}$$ ​ $$c =$$ __________
​ $s =$ __________
​ $t =$ __________

Accepted Answer

The answer of Solve the standard minimization problem using duality.
​
Minimize...

Question 18

Maximize $p = 19 x + 8 y$ subject to the following.
​ $$\begin{array} { l } 
4 x + 4 y \leq 32 \
4 x + 16 y \leq 64 \
x \geq 0 , y \geq 0
\end{array}$$ ​ $p =$ __________
​ $x =$ __________
​ $y =$ __________

Accepted Answer

The answer of Maximize \(p = 19 x + 8...

Question 19

Maximize $p = 1.2 x + 1.1 y$ over the region defined by the inequalities.   $$\left\{ \begin{array} { c } 
9 x + 8 y \leq 144 \
x + y \leq 17 \
x \geq 0 , y \geq 0
\end{array} \right\}$$

Accepted Answer

A)  20.4 
B)  18.7 
C)  19.2 
D)  19.5 
E)  18.4 
A)  20.4 
B)  18.7 
C)  19.2 
D)  19.5 
E)  18.4

Question 20

Ruff makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 200 grams of protein and at least 100 grams of fat. If chicken costs 25 cents/ounce and grain costs 1 cents/ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost 
 
__________ oz of chicken and __________ oz of grain.
 
What are the shadow costs of protein and fat 
 
__________ cent(s) per gram of protein and __________ cent(s) per gram of fat.

Accepted Answer

The answer of Ruff makes dog food out of chicken...

Maximize $p = 15 x + 14 y$ subject to the following. 2x+2y\leq8 2x+8y\leq16 x\geq0,y\geq0

Solve the linear programming problem. Maximize $P = 2 x + 8 y$ subject to $x + y \leq 4$ $2 x + y \leq 5$ $x \geq 0 , y \geq 0$

Solve the standard minimization problem using duality. Minimize $c = 4 s + t + 7 u$ subject to s+t+u \geq50 2s+t \geq25 t+u \geq25 s\geq0,t\geq0,u \geq0 $c =$ _____ $s =$ _ $t =$ $u =$ ____

Graph the solution set to the inequality. Label coordinates of all corner points (if any). $\left\{ \begin{array} { l } 2 x + y \leq 16 \\x - 2 y \leq 8\end{array} \right.$

Use the simplex method to maximize $p = 2 x + y + 3 z + w + 2 v$ subject to the following. x+y\leq3 y+z\leq1 z+w\leq3 w+v\leq6 $x \geq 0 , y \geq 0 , z \geq 0 , w \geq 0 , v \geq 0$ $p =$ $x =$ $y =$ $z =$ $w ^ { \prime } =$ $v =$

Solve the standard minimization problem using duality. Minimize $c = 0.3 s + 0.5 t$ subject to 0.2s+0.1t\geq1 0.15s+0.3t\geq1.5 10s+10t\geq80 s\geq0,t\geq0 $c =$ $s =$ $t =$ __

Minimize $c = 4 x + 7 y$ Subject to 8x+3y\geq30 2x+3y\geq12 x\geq0,y\geq0

Solve the standard minimization problem using duality. Minimize $c = 7 s + 6 t$ subject to s+t\geq12 7s+t\geq42 s\geq0,t\geq0 $c =$ $s =$ $t =$ __

Maximize $p = 19 x + 8 y$ subject to the following. 4x+4y\leq32 4x+16y\leq64 x\geq0,y\geq0 $p =$ $x =$ $y =$ __

Maximize $p = 1.2 x + 1.1 y$ over the region defined by the inequalities. $\left\{ \begin{array} { c } 9 x + 8 y \leq 144 \\x + y \leq 17 \\x \geq 0 , y \geq 0\end{array} \right\}$

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Sets and Counting

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Introduction to the Derivative

Techniques of Differentiation

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Exam 6: Linear Programming

Maximize p=15x+14yp = 15 x + 14 yp=15x+14y subject to the following. 2x+2y\leq8 2x+8y\leq16 x\geq0,y\geq0

Solve the linear programming problem. Maximize P=2x+8yP = 2 x + 8 yP=2x+8y subject to x+y≤4x + y \leq 4x+y≤4 2x+y≤52 x + y \leq 52x+y≤5 x≥0,y≥0x \geq 0 , y \geq 0x≥0,y≥0

Solve the standard minimization problem using duality. ​ Minimize c=4s+t+7uc = 4 s + t + 7 uc=4s+t+7u ​ subject to s+t+u \geq50 2s+t \geq25 t+u \geq25 s\geq0,t\geq0,u \geq0 ​ c=c =c= _________ ​ s=s =s= _________ ​ t=t =t= __________ ​ u=u =u= __________

Graph the solution set to the inequality. Label coordinates of all corner points (if any). {2x+y≤16x−2y≤8\left\{ \begin{array} { l } 2 x + y \leq 16 \\x - 2 y \leq 8\end{array} \right.{2x+y≤16x−2y≤8​

Solve the standard minimization problem using duality. ​ Minimize c=0.3s+0.5tc = 0.3 s + 0.5 tc=0.3s+0.5t ​ subject to 0.2s+0.1t\geq1 0.15s+0.3t\geq1.5 10s+10t\geq80 s\geq0,t\geq0 ​ c=c =c= __________ ​ s=s =s= __________ ​ t=t =t= __________

Minimize c=4x+7yc = 4 x + 7 yc=4x+7y Subject to 8x+3y\geq30 2x+3y\geq12 x\geq0,y\geq0

Solve the standard minimization problem using duality. ​ Minimize c=7s+6tc = 7 s + 6 tc=7s+6t ​ subject to s+t\geq12 7s+t\geq42 s\geq0,t\geq0 ​ c=c =c= __________ ​ s=s =s= __________ ​ t=t =t= __________

Maximize p=19x+8yp = 19 x + 8 yp=19x+8y subject to the following. ​ 4x+4y\leq32 4x+16y\leq64 x\geq0,y\geq0 ​ p=p =p= __________ ​ x=x =x= __________ ​ y=y =y= __________

Maximize p=1.2x+1.1yp = 1.2 x + 1.1 yp=1.2x+1.1y over the region defined by the inequalities. {9x+8y≤144x+y≤17x≥0,y≥0}\left\{ \begin{array} { c } 9 x + 8 y \leq 144 \\x + y \leq 17 \\x \geq 0 , y \geq 0\end{array} \right\}⎩⎨⎧​9x+8y≤144x+y≤17x≥0,y≥0​⎭⎬⎫​

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Sets and Counting

Probability

Random Variables and Statistics

Introduction to the Derivative

Techniques of Differentiation

Applications of the Derivative

The Integral

Further Integration Techniques and Applications of the Integral

Functions of Several Variables

Trigonometric Models

Filters

Maximize $p = 15 x + 14 y$ subject to the following. 2x+2y\leq8 2x+8y\leq16 x\geq0,y\geq0

Solve the linear programming problem. Maximize $P = 2 x + 8 y$ subject to $x + y \leq 4$ $2 x + y \leq 5$ $x \geq 0 , y \geq 0$

Solve the standard minimization problem using duality. Minimize $c = 4 s + t + 7 u$ subject to s+t+u \geq50 2s+t \geq25 t+u \geq25 s\geq0,t\geq0,u \geq0 $c =$ _____ $s =$ _ $t =$ $u =$ ____

Graph the solution set to the inequality. Label coordinates of all corner points (if any). $\left\{ \begin{array} { l } 2 x + y \leq 16 \\x - 2 y \leq 8\end{array} \right.$

Solve the standard minimization problem using duality. Minimize $c = 0.3 s + 0.5 t$ subject to 0.2s+0.1t\geq1 0.15s+0.3t\geq1.5 10s+10t\geq80 s\geq0,t\geq0 $c =$ $s =$ $t =$ __

Minimize $c = 4 x + 7 y$ Subject to 8x+3y\geq30 2x+3y\geq12 x\geq0,y\geq0

Solve the standard minimization problem using duality. Minimize $c = 7 s + 6 t$ subject to s+t\geq12 7s+t\geq42 s\geq0,t\geq0 $c =$ $s =$ $t =$ __

Maximize $p = 19 x + 8 y$ subject to the following. 4x+4y\leq32 4x+16y\leq64 x\geq0,y\geq0 $p =$ $x =$ $y =$ __

Maximize $p = 1.2 x + 1.1 y$ over the region defined by the inequalities. $\left\{ \begin{array} { c } 9 x + 8 y \leq 144 \\x + y \leq 17 \\x \geq 0 , y \geq 0\end{array} \right\}$