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.

hi there!

first we need to find out how much they are adding to each number to make the next number.

ex: 4 + 4 = 8, 8 + 8 = 16, and so on.

so continuing with 32,

32 + 32 = 64

64 + 64 = 128

128 + 128 = 256

256 + 256 = 512

4, 8, 16, 32, 64, 128, 256, 512

the 8th term of the sequence will be 512

hope this ! ~

answer: 10 turtles

step-by-step explanation: as turtles have four legs so 4.10=40 therefore birds are 4 which sue counted