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

Answer:

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

Step-by-step explanation:

using formula:

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

$\to revenue = 70 \times 4\pi r^2 = 280\pi r^2\\\\\to cost = 800 \times \frac{4}{3} \pi r^3 = \frac{3200}{3} \pi r^3$

so,

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

$\to \pi r(560 -3200r)=0\\\\ \to (560 -3200r)=0\\\\\to 3200r=560\\\\\to r=\frac{560}{3200}\\\\\to r=\frac{56}{320}\\\\\to r=\frac{7}{40}\\\\\to r=0.175 \ ft$