# Max-Min Problems for Precalculus Students

### Professor Rothchild, MAT172, Lehman College, CUNY

1.The sum of two numbers is 34.
a) Find the largest possible product of such numbers.
b) What would be the largest possible product if the sum of the two numbers were k?
2.Sixty meters of fencing are to be used to enclose a rectangular garden.
a) Find the dimensions that will give the maximum area.
b) What would be the maximum area if k feet of fencing were used?
3.Forty feet of fencing are to enclose three sides of a rectangular patio. The fourth side
of the patio is a side of the house, and requires no fencing.
a) Find the maximum area of the patio.
b) What would be the maximum area if k feet of fencing were used?
4. A grasshopper is h(t) = 8t - 4t2 inches above the ground t seconds after it jumps.
a) How high does the grasshopper jump?
b) How long is the grasshopper in the air?
c) What relation does your answers to (a) and (b) have to the domain and the range of h?
5.Find the point on the graph of y = 2x + 10 that is closest to the origin.

6. Find the point on the graph of y = ( x2 - 1 )1/2 that is closest to the point (0, 1).

7. A motel has 20 rooms. If the manager charges \$60 per room per night, all the rooms
will be rented. For each \$5 increase, one less room will be rented.

a) How much rent should be charged to maximize revenue?
b) How many rooms are left empty to achieve this maximum?
8. If a publisher charges \$50 for a textbook, she will sell 8000 copies.
For each \$1 she raises the price, she sells 100 fewer copies,
and for each \$1 she lowers the price, she sells 100 more copies.
a) How much should she charge to maximize revenue?
b) How many books will she sell at that rate?
9. I own 200 shares of Flybinite Enterprises that are worth \$75 apiece. Each year I hold them,
I get 5 free shares but the value of each share I have decreases by \$2.
a) How many years should I hold these shares to maximize their total value?
b) Why did you need to consider the domain of your function to answer (a) correctly?
10.The revenue from the sale of x items is R(x) = 800x - 2x2 dollars and the cost to
produce these x items is C(x) = 2x2 + 1000 dollars.
a) How many items should be produced to minimize cost?
b) How many items should be produced to maximize revenue?
c) How many items should be produced to maximize profit?
d) If profit is to be a maximum, what should be the selling price of each item?