Optimization Problem

Question

A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 96 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)

x=

y=

Answer 1

If the package has a combined length and girth of 96 inches. We can determine that:

4x + y = 96

We are also assuming that the cross section is square, so:

V = x²y

Let's solve for y in the first equation and plug it in to the second equation:

y = 96 - 4x

V = x²(96 - 4x)

V = 96x² - 4x³

In order to solve for the maximum is when V' = 0

V' = 192x - 12x² = 0

x(192 - 12x) = 0

Therefore, we have 2 solutions,

x = 0

192 - 12x = 0

12x = 192

x = 16

x = 0 is when the package is a minimum, so the maximum occurs when x = 16. Now let's solve for y

y = 96 - 4x = 96 - 4(16) = 32

The answers are:

x = 16 inches

y = 32 inches

Answer 2

h+4*a=96, here h is the length, and a is the length of side of square,

the volume goes as V=h*a**2.

If we use the relations between a and h above, we can get h=96-4a

Put this relation into volume, we get the volume be

V=(96-4a)a**2.

At this point, it has become clear that we need to find the maximum of this polynomial.

We need to find a point where the first derivate of V vanishes and also need to make sure the second derivation being positive.

V'=(96-4a)*2a-4a**2=2a(96-6a)=0 ==>>a(16-a)=0 >>a=0,16

The only possiblility is a=16.

But let's check to make sure it's indeed maximum

V''=192-24a, and V''(a=16)=192-24*16<0.

SO the answer is a=16, length h=96-4a=96-64=32