Question

A certain magical substance that is used to make solid magical spheres costs $500 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?Justify why it is a maximum rather than a minimum.

Answer 1

r= 0.08 ft

Explanation:

C= $500 per cubic foot

S=$20 per square foot of surface area

We know that

Profit = Selling price - Manufacturing price

P = S - C

Lets take r is the radius of sphere

Surface area ,A= 4πrr²

Volume ,V= 4πr³/3

P = S - C

P = 20 x 4πr² - 500 x 4πr³/3

For maximize the profit

dP/dr = 0

P = 20 x 4πr² - 500 x 4πr³/3

dP/dr = 160 πr - 2000 πr²

160 πr - 2000 πr² = 0

160 - 2000 r = 0

r= 0.08 ft

d²P/dr² = - 4000 πr

d²P/dr² is negative at all value of r that is why at r=0.08 ft ,it is maximum.