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
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.
Show 1/4 of a set of 8 counters
To maximize profit per sphere, we need to find the size (volume) of the magical sphere that will yield the maximum profit.
Let's denote:
- V as the volume of the magical sphere (in cubic feet)
- S as the surface area of the magical sphere (in square feet)
- C as the cost of the magical substance per cubic foot ($800)
- P as the selling price per square foot of surface area ($20)
- N as the number of spheres produced
We know that the volume of a sphere is given by the formula:
V = (4/3) * π * r^3
And the surface area of a sphere is given by the formula:
S = 4 * π * r^2
We can express the radius (r) in terms of the volume (V) as:
r = (V * (3/4) * (1/π))^(1/3)
Substituting this value of r in the surface area formula, we get:
S = 4 * π * [(V * (3/4) * (1/π))^(1/3)]^2
Simplified as:
S = 4 * π * [(3 * V) / (4 * π^(2/3))]
Now, we can express the profit per sphere as:
Profit = Selling Price - Cost
Profit = P * S - C * V
Profit = (20 * 4 * π * [(3 * V) / (4 * π^(2/3))]) - (800 * V)
Profit = (80 * π * (3 * V) * (π^(-2/3))) - (800 * V)
Profit = [(240 * π * V) / (π^(2/3))] - (800 * V)
To maximize the profit per sphere, we need to find the value of V that maximizes this expression. We can do this by differentiating the expression with respect to V and setting it equal to zero:
d(Profit)/dV = [(240 * π) / (π^(2/3))] - 800 = 0
We can solve this equation to find the value of π that maximizes the profit per sphere. However, it seems there is a mistake in the expression. I apologize for the inconvenience.
To determine the size of the magical sphere that will maximize your profit, we need to find the surface area that will generate the highest profit and calculate the corresponding dimensions.
Let's start by considering the cost of manufacturing a magical sphere. The cost of the magical substance is given as $800 per cubic foot, but we need to find the cost per square foot of surface area. To do that, we need to convert the cost per cubic foot to cost per square foot.
First, we need to find the volume of the sphere in terms of its radius, which we'll call r. The formula for the volume of a sphere is:
V = (4/3)πr^3
Since the surface area of a sphere depends only on its radius, we can find the surface area formula by differentiating the volume formula with respect to the radius:
dV/dr = 4πr^2
Now, let's calculate the cost per cubic foot in terms of the radius. We'll divide the cost per cubic foot of the magical substance by the volume formula:
Cost per cubic foot = $800
Cost per cubic foot = $800 / V
Cost per cubic foot = $800 / [(4/3)πr^3]
Next, let's convert the cost per cubic foot to cost per square foot. Since we know that the surface area of a sphere is given by the formula:
A = 4πr^2
We can divide the cost per cubic foot by this surface area formula to get the cost per square foot:
Cost per square foot = Cost per cubic foot / A
Cost per square foot = [$800 / [(4/3)πr^3]] / [4πr^2]
Cost per square foot = (3/4) * [$800 / r]
Now, let's calculate the profit per sphere. The selling price for each square foot of surface area is $20:
Profit per sphere = Selling price per square foot - Cost per square foot
Profit per sphere = $20 - [(3/4) * ($800 / r)]
To maximize the profit per sphere, we need to find the value of r that maximizes this expression. To do that, we'll differentiate the profit expression with respect to r and set it equal to zero:
d(Profit)/dr = 0
d/dx [$20 - (3/4) * ($800 / r)] = 0
Simplifying this, we get:
d/dx [$20 - ($2,400 / r)] = 0
-($2,400) / r^2 = 0
Solving for r, we get:
r^2 = ($2,400) / 0
r^2 = infinity
From this, we can see that the derivative is undefined. This means that there is no critical point for the profit function. Therefore, there is no specific size of sphere that will maximize profit.
In summary, based on the given information, there is no specific size of magical sphere that will maximize your profit per sphere. The profit per sphere will depend on the surface area and the corresponding cost of manufacturing.