Name______Calculus

Optimization Problems #2

  1. What positive number exceeds its square by the greatest amount?
  1. A net enclosure for golf practice is open at one end. The volume of the enclosure is cubic meters. Find the dimensions that require the smallest amount of netting.
  1. A rectangular page is to contain 36 square inches of print. The margins at the top and bottom and on each side are to be inches. Find the dimensions of the page that will minimize the amount of paper used.
  1. A rectangle is bound by the x- and y-axes and the graph of . What length and width should the rectangle have so that its area is a maximum?
  1. You want to minimize the sum of a positive number and twice its reciprocal. What number does this, and how small can this sum be?
  1. You are designing a soft drink container that has the shape of a right circular cylinder. The container is supposed to hold 12 fluid ounces (1 fluid ounce is approximately 1.80469 cubic inches). Find the dimensions that will use a minimum amount of construction material.
  1. Find the point on the graph of the function that is closest to the given point.
  1. There are currently 50 apple trees in an orchard. Each tree produces 800 apples per year. The farmer is considering planting more trees in order to get more apples. Unfortunately, for each additional tree planted in the orchard, the output for all trees drops by 5 apples.

a. What is the total current output (per year)?
b. If the farmer plants 10 more trees, what will the output be?
c. If the farmer plants x more trees, what will the output be?
d. How many trees should be added in order to maximize the total output? /
  1. A farmer has 500 bushels of oranges ready for market. The market price is currently $6 per bushel. She thinks the price will rise 20 cents per bushel each week. But each week, 10 bushels of oranges go bad. How many weeks should she wait before selling the oranges if she wants to maximize the total dollars she gets?
  1. A running track is a rectangle with a semicircle on each end. The perimeter of the track must be 200 meters. You want to maximize the area of the rectangular portion (for playing fields). What are the dimensions of the largest possible rectangle? [Hint: define your variables as the length and width of the rectangle.]

1. so x=1/2

5. so and the sum is

8. a. 50*800=40,000b. 60*750=45,000c. (50+x)(800-5x)d. x=55 more trees…55,125 apples

9. dollars = bushels*price ; if she waits x weeks then dollars=(500-10x)(6+0.2x)  10 weeks

10. x=length of rectangle and y=width; track perimeter is

Maximize Area ; answer is x=50 and y=