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3 December, 07:57

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?

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  1. 3 December, 08:26
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    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.
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