Find the dimensions (length × width × height) of the container with greatest volume.

Question

A container with square base, vertical sides, and open top is to be made from 1200 square feet of material. Find the dimensions (length × width × height) of the container with greatest volume.

Answer 1

surface area = 1200 ft^2

V= (x^2)h where h= height and x=side of the square bottom

SA = 1200 = x^2 + 4xh = bottom area plus 4 times a side area

4xh = 1200-x^2

h = 300/x - x/4

V(x) = (x^2)(300/x -x/4) = 300x -(1/4)x^3

take the derivative and set equal to zero

V'(x) = 300-(3/4)x^2 = 0

(3/4)x^2 =300

(1/4)x^2 = 100

x^2 = 400

x=20 feet = side of the square bottom

h=300/20 -20/4 = 15-5 = 10 feet

V=10(20)^2 = 4,000 ft^3= max volume when material is limited to 1200 square feet and the top is open

dimensions are 20 ft by 20 ft by 10 ft