Optimization problem?

Question

I  am really confuse to this problem. I need some Idea how to start to solve this Optimization problem.

llaffer · Accepted Answer

Starting with the volume of a box with a square base:

V = s²h

We know this is to be fixed at 32 ft³, so:

32 = s²h

Let's solve this for h in terms of s as we'll need it later:

32 / s² = h

You want to find the minimum surface area of the box with an open-top, so we'll start with that equation:

A = 4sh + s²

We can substitute the expression in terms of "s" in for h and simplify:

A = 4s * 32 / s² + s²
A = 128s / s² + s²
A = 128 / s + s²

The "s" that gives a minimum "A" can be found by solving for the zero of the first derivative:

dA/ds = -128 / s² + 2s
0 = -128 / s² + 2s

Multiply both sides by s²:

0 = -128 + 2s³
128 = 2s³
64 = s³
4 = s

Now that we have this we can solve for A.  We can also solve for h, but we aren't asked for that:

A = 128 / s + s²
A = 128 / 4 + 4²
A = 32 + 16
A = 48 ft²

That's the minimum surface area for this container.