Finding the max volume of a box

Question

A company wants to manufacture a box for which the rectangular bottom has a perimeter of 64 cm. The height is a fixed 12 cm. Use an equation to algebraically find the maximum volume of the box.

Answer 1

2(a+b)=64, b=32-a

V=12ab=12a(32-a)

V'_a=dV/da=12(32-a)-12a=12*32-24a=0, --> a=16.

V''_a=-24<0 --> a=16  is maximum.

So,  V=3072 cm³

Answer 2

since the height is fixed, you only need to consider the base.

it's well known that the maximum area of a rectangle of fixed perimeter occurs when the figure is a square.

 

SOLUTION: 16 cm by 16 cm =base, height 12 (fixed).

Base Area =64 sq cm, base perimeter = 4(16 = 64 cm.