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

Answer:

x  =  0.1428 ft

Step-by-step 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