Calculus

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 $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

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  1. profit = revenue-cost
    revenue = 20*4πr^2 = 80πr^2
    cost = 800*4/3 πr^3 = 3200/3 πr^3
    so,
    profit = 80πr^2 - 3200/3 πr^3
    max profit when
    160πr - 3200πr^2 = 0
    160πr(1-20r) = 0
    r = 1/20 ft

    looks like marbles make the most money.

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