Question

(1 point) 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 $70 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

The answer is "
`r=0.175 \ ft`"

Explanation:

using formula:

`\to \text{profit = revenue-cost}`

`\to revenue = 70 * 4\pi r^2 = 280\pi r^2\\\\\to cost = 800 * (4)/(3) \pi r^3 = (3200)/(3) \pi r^3`

so,

`\to profit (p)= 280\pi r^2 - (3200)/(3) \pi r^3\\\\\text{max profit when}\\\\\to 560 \pi r-3200 \pi r^2=0`

`\to \pi r(560 -3200r)=0\\\\ \to (560 -3200r)=0\\\\\to 3200r=560\\\\\to r=(560)/(3200)\\\\\to r=(56)/(320)\\\\\to r=(7)/(40)\\\\\to r=0.175 \ ft`