A certain magical substance that is used to make solid magical spheres

Question

A certain magical substance that is used to make solid magical spheres costs 700\$ per cubic foot. The power of a magical sphere depends on its surface area, and a magical sphere can be sold for 40\$ per square foot of surface area.

If you are manufacturing such a sphere, what size should you make them to maximize your profit per sphere?

Answer 1

Let:

1. $r$ be the radius of a magical sphere.
2. $p(r)$ be the profit per magical sphere.
3. $s(r)$ be the sale price of a magical sphere
4. $c(r)$ be the cost of making one magical sphere.

$\Rightarrow p(r)=s(r)-c(r)$

Thus, according to the question and the known formulas for the surface area and volume of a sphere:

$s(r) = 40\times4\pi r^2=160\pi r^2$

$c(r) = 700\times\frac{4}{3}\pi r^3=\frac{2800}{3}\pi r^3$

And then:

$p(r)=160\pi r^2-\frac{2800}{3}\pi r^3$

Now, for finding the maximum profit:

$p'(r)=320\pi r-2800\pi r^2=0$

Solving for $r$ yields: $$r=\frac{4}{35}$$