storage bin with rectangular base and length is twice the width of the base

Question

a storage bin has a rectangular base and open top. the length of the base is twice the width of the base, the manufacturer wants to use only 20 square feet of material to build the container.

1. find a formula for the volume of the bin
2. use info from the problem to write your formula in terms of one variable
3. find the dimensions of the bin which maximize the volume.

Answer 1

1. V= l·w·h = 2w²h
2. 20 = lw +2lh + 2wh; l = 2w; 20 = 2w² + 6wh; h = (10 - w²)/(3w)
3. V = 2w²(10 - w²)/(3w) = 2/3(10w - w³) ;

dV/dw = 2/3(10 - 3w²) = 0; 3w² = 10; w = √10/3

V'' = - 4w < 0, so at w = √10/3 we have maxinum volume.

w = √10/3 = 1.826 ft; l = 2√10/3 = 3.6515 ft; ; h = (10 - 10/3)/(3√10/3) = 20/(9√10/3) = 2/3√10/3 = 1.2172ft