Question

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?

Answer 1

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

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:

⇒
`P(r)=(unit\ cost)\ (SA) - (unit\ cost)\ (Volume)`

⇒
`P(r)=(30)\ (4 \pi r^2) - (500)\ ((4\pi r^3)/(3) )`

⇒
`P(r)=(120)\ (2\pi r^2) - ((500* 4)/(3) )\ \pi r^3`

⇒
`P(r)=(120)\ (\pi r^2) - ((2000)/(3) )\ \pi r^3`

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

⇒
`P(r)'=(120)\ 2\pi r - ((2000)/(3) )\ 3\pi r^2`

⇒
`P(r)'=(240)\ \pi r - (2000)\ \pi r^2`

Finding r values :

⇒
`(240)\ \pi r - (2000)\ \pi r^2 =0`

Dividing both sides with 240π .

⇒
`r-(25)/(3) r^2 =0` ⇒
`r(1-(25)/(3) r) =0`

⇒
`r=0` and
`r=(3)/(25)`

To find maxima value the double differentiation is :

⇒
`P(r)'=(240)\ \pi r - (2000)\ \pi r^2` ...first derivative

Double differentiating :

⇒
`P(r)''=(240\pi) - (2000\pi)\ 2(r)` ...second derivative

⇒
`P(r)''=(240\pi) - (4000\pi)\ (r)`

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

⇒
`1 - (50\pi r)/(3)`

⇒
`1 - (50* \pi* 3 )/(3* 25)`

⇒
`-5.28 < 0`

It passed the double differentiation test.

Extra work :

Thus:

⇒
`P(r)=(120)\ (\pi r^2) - ((2000)/(3) )\ \pi r^3`

⇒
`P(r)=(120)* (\pi ((3)/(25) )^2) - ((2000)/(3) )* \pi ((3)/(25) )^3`

⇒
`P(r) =1.8095`

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