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 $50 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

x = 0.1428 ft

Explanation:

Let x be radius of sphere

Cost of magical sphere is

700 $ per ft³ and the volume of the sphere of radius x is V = (4/3)*π*x³

700*(4/3)*π*x³

The selling price is :

Area of the sphere : 4*π*x² selling price = 50*4*π*x²

Then my profit = selling price - cost

P(x) = 200πx² - 700 (4/3)πx³

Taking derivative both sides

P´(x) = 400πx - 2800πx²

P´(x) = 0 ⇒ 400πx - 2800πx² = 0

xπ ( 400 - 2800x ) = 0 x ( 400 - 2800x ) = 0

x = 0

And 400 - 2800x = 0 x = 400/2800 x = 0.1428 ft