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:

⇒

⇒

⇒

⇒

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

⇒

⇒

Finding r values :

⇒  

Dividing both sides with 240π .

⇒    ⇒

⇒ and  

To find maxima value the double differentiation is :

⇒     ...first derivative

Double differentiating :

⇒    ...second derivative

⇒

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

⇒  

⇒

⇒

It passed the double differentiation test.

Extra work :

Thus:

⇒

⇒

⇒

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

i think it is 70

step-by-step explanation:

answer: the angel abc is 30°,and the abc is a triangle where the angel bac is 90°

so ac is the half of 24=12ft,and from the pythagorean theorem we find ba

ba=√bc^2-ac^2=√576-144=√432=12√3ft

so bac is 12+12√3=12(1+√3)ft