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 $30 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?

the sphere should be constructed with a radius R = 0.075 ft in order to maximise the profit

Step-by-step explanation:

since the profit function P is

P = $30/ft² * A - $800/ft³ * V = a*A + b*V

where A and V are the area and the volume of the sphere respectively. Then

A = 4*π*R²

and

V = 4/3*π*R³

where R is the radius

replacing in P

P = a*A + b*V = 4*π*a * R² - 4/3*π*b*R³ =

the maximum is found where the derivative of P with respect to R is equal to 0, therefore:

dP/dR = 8*π*a * R - 4*π*b*R² = 0

then

8*π*a * R - 4*π*b*R² = 0

4*π*R * (2*a - b*R) = 0

since R>0

2*a - b*R=0

R = 2*a/b

replacing values

R = 2*a/b = 2*$30/ft² / $800/ft³ = 0.075 ft

thus the sphere should be constructed with a radius R = 0.075 ft in order to maximise the profit.