Calculus 1 - Optimization

Question

1. You are to make a box without a lid from a 100 cm by 35 cm piece of cardboard. What are the dimensions that will maximize the volume? What is the maximum volume?

Answer 1

Surface Area = A = 100x35 = 3500

A= 4HB + B^2 where H=height and B = side of the base of the box

4HB = 3500-B^2

H = (3500-B^2)/4B = 875/B-B/4

V=HB^2 = (875/B-B/4)B^2 = 875B - B^3/4

V' = 875 -3B^2/4 = 0

B = about 34.2 = a side of the base of the box

H = 875/34.2 -34.2/4 = 25.6 -8.5 = 17.1

V = HB^2 = 17.1(34.2)^2 = 19,924.6 in^3