A certain magical substance that is used to make solid magical spheres costs per cubic foot. The power of a magical sphere depends on its surface area, and a magical sphere can be sold for 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?
Thank you very much for your help. =)
Does the price depend on the magical power? What is the relations between:
magic power and price
cost / cubic foot
selling price / sq. ft
If the magic power doesn't count, then make the spheres as physically possible.
To determine the size of the magical sphere that will maximize your profit per sphere, we need to consider the cost of the magical substance and the selling price based on the surface area.
Let's denote the cost of the magical substance as C dollars per cubic foot and the selling price as P dollars per square foot of surface area.
First, we need to express the volume of the sphere in terms of its surface area. The volume of a sphere can be calculated using the formula:
V = (4/3)πr^3
Since we are looking for a relationship between volume and surface area, we need to express the radius in terms of the surface area. The surface area of a sphere can be calculated using the formula:
A = 4πr^2
Rearranging the surface area formula, we get:
r^2 = A / (4π)
Now, substitute this value of r^2 into the volume formula:
V = (4/3)π[(A / (4π))^3/2]
Simplifying the volume formula, we get:
V = (√3/36π^2)A^(3/2)
Next, we can calculate the cost of the magical substance for a given volume. Since the cost is given per cubic foot, we multiply the volume by the cost per cubic foot:
Cost = C * V
Substitute the volume formula into the cost formula:
Cost = C * (√3/36π^2)A^(3/2)
Now, let's calculate the profit for a given surface area. The profit is obtained by subtracting the cost from the selling price:
Profit = P * A - Cost
Substitute the cost formula into the profit formula:
Profit = P * A - (C * (√3/36π^2)A^(3/2))
To maximize the profit per sphere, we take the derivative of the profit formula with respect to A, and set it equal to zero:
d(Profit) / dA = 0
Differentiating the profit formula, we get:
P - (C * (√3/24π^2) * (3/2) * A^(1/2)) = 0
Simplifying the equation:
P = (C * (√3/24π^2) * (3/2) * A^(1/2))
Now, solve for A to find the surface area that maximizes profit:
A^(1/2) = (2P) / (C * (√3/24π^2) * (3/2))
A = [(2P) / (C * (√3/24π^2) * (3/2))]^2
Simplify further:
A = (16P^2 * 36π^2) / (3C^2)
Hence, to maximize profit per sphere, you should manufacture spheres with a surface area of (16P^2 * 36π^2) / (3C^2) square feet.