CALCULUS 1 – OPTIMIZATION REVIEW HW
NAME:
OPTIMIZATION – DAY 4
DATE:
1.
Find two positive numbers whose sum of twice the first and four times the second is 80 and
whose product is as large as possible. What is the product?
2.
A trough is made from a rectangular piece of metal 32 inches wide by turning up the sides at right
angles at the base. What are the dimensions of the trough with the greatest carrying capacity? What is
that capacity?
3.
A rancher has 800 meters of fence to enclose 6 adjacent rectangular fields. Determine the
dimensions of each field that would maximize the area of the fields. What is the area of each field?
4.
A poster will contain 100 square inches of text with a 3 inch margin on top and bottom and a 2
inch margin on the left and right. What overall dimensions would minimize the paper used for the poster?
5.
A manufacturer wants to make an open box. She will make it from a 4 foot by 6 foot piece of
material by cutting equal squares from each corner and turning up the sides. Find the dimensions and
the volume of the largest box that can be made.
6.
A company wants to make a box with a lid that has a square base and a volume of 17,576 cubic
inches. Find the dimensions and the surface area of the box that uses the least amount of material.
7.
A designer wants to make a cylinder that holds 500 cubic centimeters of liquid. Find the
dimensions and the surface area of the cylinder that uses the least amount of material.
8.
Find the point on the graph of 9𝑥 + 4𝑦 2 = 36 that is nearest to (−2, 0). What is the distance
between the two points?
9.
A right cone is inscribed in a sphere of radius 10. Find the dimensions of the cone that has the
greatest volume. What is the maximum volume?
10.
A potato grower can ship 6 tons of potatoes at a profit of $20 per ton if he ships on July 1st. If he
waits, he can add 3 tons per week to his shipment but his profit would be reduced by $2 per ton per
week. On what date should he ship the potatoes? How much profit will he make?