A certain magical substance that is used to make solid magical spheres costs $\$800$ per cubic foot. The power of a magical sphere depends on its surface area, and a magical sphere can be sold for $\$20$ 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?
Hi im really struggling with this one. Can someone please show how to to complete question.
Note: the formulas for the volume and surface area of a sphere of radius $r$ are:
$$V= (4 \pi/3)r^3, ~~ A=(4 \pi)(r^2)$$
Let $p$ be the profit, then
$$p(r) = 20 \left(4 \pi r^2\right)-\frac{800}{3} \left(4 \pi r^3\right) = 80 \pi r^2-\frac{3200 \pi r^3}{3}$$
A plot of the profit function shows:
$p'=160 \pi r-3200 \pi r^2 = 0 \implies r = 0, r = \frac{1}{20}$ ($r \ne 0$)
$p'' = 160 \pi -6400 \pi r \rightarrow p''\left(\frac{1}{20}\right) = -160 \pi \lt 0$
$r = \frac{1}{20}$ ft gives a maximum profit, thus:
$$p\left(\frac{1}{20}\right) = \frac{\pi }{15} = 0.20944$$