Certain magical substance that is used to make solid magical spheres costs $500 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?

The value of r to have maximum profit is 3/25 ft

Step-by-step explanation:

To find:

The size of the sphere so that the profit can be maximized.

Manufacturing cost of the solid sphere = $500/ ft^3

Selling price of sphere (on surface area) = $30 / ft^2

We see that the manufacturing cost dealt with he volume of the sphere where as the selling price dealt with the surface area.

So,

To maximize the profit (P) .

We can say that:

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⇒

⇒

Differentiate "" and find the "" value then double differentiate "", plug the "" values from to find the minimum and maximum values.

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Finding r values :

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Dividing both sides with 240π .

⇒    ⇒

⇒ and  

To find maxima value the double differentiation is :

⇒     ...first derivative

Double differentiating :

⇒    ...second derivative

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Test the value r = 3/25 dividing both sides with 240π

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It passed the double differentiation test.

Extra work :

Thus:

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Finally r =3/25 ft that will maximize the profit of the manufacturing company.

srry idk

step-by-step explanation:

to find the total number of combinations, multiply the quantity of each item by each other:

2 pairs of shoes x 3 pairs of shorts = 6 different combinations.