Question

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?

Answer 1

Answer:

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.