1. Of all numbers that add to 74, find the pair that has the maximum product. That is, maximize the objective function Q=xy subject to the constraint x+y=74.

The values of x and y that have the maximum product are x= __ and y=__

2. Maximize B=5xy^2, where x and y are positive numbers such that x+y^2=9.

The maximum value of B is __

3. Minimize Q=5x^2 + 2y^2, where x+y=7.

Write the objective function in terms of x.

Q= __

4. Minimize Q=2x^2+5y^2, where x+y=7.

x= __

y= __

5. Maximize Q=xy, where x and y are positive numbers such that x+6y^2=8.

Write the objective function in terms of y. Q= __

6. Maximize Q=xy, where x and y are positive numbers such that x+(20/3)y^2=5.

The maximum value of Q is __ and occurs at x=enter your response here and y=__

7. A lifeguard needs to rope off a rectangular swimming area in front of Long Lake Beach, using

1400 yd of rope and floats. What dimensions of the rectangle will maximize the area? What is the maximum area? (Note that the shoreline is one side of the rectangle.)

8. Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 35 in. by 19 in. by cutting congruent squares from the corners and folding

up the sides. Then find the volume.

The dimensions of box of maximum volume are __

9. Drum Tight Containers is designing an open-top, square-based, rectangular box that will have a volume of 13.5 in.3. What dimensions will minimize surface area? What is the minimum surface area?

10. A waste management company is designing a rectangular construction dumpster that will be twice as long as it is wide and must hold 18 yd^3 of debris. Find the dimensions of the dumpster that will minimize its surface area.

Write the surface area formula in terms of the width, x. Assume the dumpster has an open top.

SA= __

11. Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, R(x), and cost, C(x), of producing x units are in dollars. R(x)=50x0.5x^2, C(x)=6x+ 30

12. Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue,

R(x), and cost, C(x), of producing x units are in dollars.

R(x)=4x,

C(x)=0.01x^2+0.1x+4

13. Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, R(x), and cost, C(x), are in thousands of dollars, and x is in thousands of units.

R(x)=8x2x2,

C(x)=x^3-3x^2+4x+1

14. Raggs, Ltd. a clothing firm, determines that in order to sell x suits, the price per suit must be

p=1800.5x. It also determines that the total cost of producing x suits is given by

C(x)=5000+0.75x2.

a) Find the total revenue,

R(x).

b) Find the total profit,

P(x).

c) How many suits must the company produce and sell in order to maximize profit?

d) What is the maximum profit?

e) What price per suit must be charged in order to maximize profit?

15. A university is trying to determine what price to charge for tickets to football games. At a price of $30 per ticket, attendance averages 40,000 people per game. Every decrease of

$5 adds 10,000 people to the average number. Every person at the game spends an average of $5.00 on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price?

16. An apple farm yields an average of 41 bushels of apples per tree when 21

trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield?

17. A car rental agency rents 210 cars per day at a rate of $30 per day. For each $1

increase in rate, 5 fewer cars are rented. At what rate should the cars be rented to produce the maximum income? What is the maximum income?

18. A rectangular box with a volume of 720 ft^3 is to be constructed with a square base and top. The cost per square foot for the bottom is 20, for the top is 15, and for the sides is 1.5.

What dimensions will minimize the cost?

19. A rectangular page in a text (with width x and length y) has an area of 18 in.^2, top and bottom margins set at 12 in., and left and right margins set at 1 in.

The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?

20. A travel agency offers a package for $2000 per person if they have 30 or less sign up. For each person in excess of 30, the agency reduces the price by

$20 for every traveler.

(a)Complete the table to explore the relationship between price and revenue numerically.(b)Generalize the numerical pattern in the table to write formulas for the price as a function of number of travelers and revenue as a function of travelers.

(c)What number of travelers must sign up to produce the maximum revenue? What is the maximum revenue and what will each passenger pay?

(d)It will cost the travel agency

C(n)=9000+200n

for this trip if they have n travelers. Determine the number of travelers needed to produce the maximum profit. What is the maximum profit and what will each passenger pay?

Number of Travelers, n	Price per person, p(n)	Revenue R(n)
30	$_____	$_____
31	$_____	$_____
32	$_____	$_____
40	$ _____	$_____

21. An amusement park charges $10 for admission. On average, 25,000

people visit the park each day. Suppose that for each $1 increase in the entrance price, the park loses 250 daily customers.

(a)Complete the table to explore the relationship between price and revenue numerically.(b)Generalize the numerical pattern in the table to write a formulas for the number of patrons as a function of price and the revenue as a function of price.

(c)What price should be charged to produce the maximum revenue? What is the maximum revenue? What number of patrons will visit the park?

Admission price, p	Attendance, N(p)	Revenue R(p)
$10	$____	$____
$11	$____	$____
$12	$____	$____