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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?

User Boomboxboy
by
9.2k points

1 Answer

4 votes

Answer:

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.

User Victor Santizo
by
7.7k points
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