Calculus find Width Height and Length(Box with no top)

Question

You have 150 square feet of material to build 

a rectangular box.

The length is the same as the width and 

there is no top.

What values for W and L and H will create a box that 

encloses the maximum volume using the material you have?

W=

L=

H=

Answer 1

Let V=s²h where s= side and h=height of the box. So 150= s² + 4sh and h= (150 -s²)/4s. So the volume now becomes V= s²(150 - s²)/4s. V= s( 150- s²)/4. Taking the derivative dV/ds =(150-s²)/4 -s²/2 = 150/4 -3s²/4. The second derivative gives -3s/2 so this is a maximum. Solving for s gives 3s²= 150 and s= 5√2 and

h= (150-50)/4•5√2 h= 100/20√2 = 5/√2 =2.5√2. So L=W= 5√2 and H= 2.5√2.