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

Answer:

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:

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

⇒ $P(r)=(30)\ (4 \pi r^2) - (500)\ (\frac{4\pi r^3}{3} )$

⇒ $P(r)=(120)\ (2\pi r^2) - (\frac{500\times 4}{3} )\ \pi r^3$

⇒ $P(r)=(120)\ (\pi r^2) - (\frac{2000}{3} )\ \pi r^3$

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

⇒ $P(r)'=(120)\ 2\pi r - (\frac{2000}{3} )\ 3\pi r^2$

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

Finding r values :

⇒ $(240)\ \pi r - (2000)\ \pi r^2 =0$  

Dividing both sides with 240π .

⇒ $r-\frac{25}{3} r^2 =0$    ⇒ $r(1-\frac{25}{3} r) =0$

⇒ $r=0$ and $r=\frac{3}{25}$  

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

⇒  $1 - \frac{50\pi r}{3}$

⇒ $1 - \frac{50\times \pi\times 3 }{3\times 25}$

⇒ $-5.28 < 0$

It passed the double differentiation test.

Extra work :

Thus:

⇒ $P(r)=(120)\ (\pi r^2) - (\frac{2000}{3} )\ \pi r^3$

⇒ $P(r)=(120)\times (\pi (\frac{3}{25} )^2) - (\frac{2000}{3} )\times \pi (\frac{3}{25} )^3$

⇒ $P(r) =1.8095$

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