# Thread: Volume and Surface area derivative problem

1. ## Volume and Surface area derivative problem

Hi I was hoping for some help on this 3 step problem:

a) Given Surface area of sphere, A = 4*pi*r^2 and Volume = 4/3*pi* r^3, solve for V in terms of A.

b) Find dV/dA.

c) The volume of a ball of radius 4in is increased by .5 in^3. Estimate the increase in the
area.

I get stuck at part c...

a) I solved for r in area formula and substituted in the volume formula to get:
V= A/3* sqrt(A/4*pi)

b) dV/dA = Sqrt(x)/4*sqrt(pi) *thanks to wolfram alpha

c) My first thought was to find volume of a ball with radius 4, add .5 then solve for the new radius. Then plug that into the surface area formula and see what I get. However I feel there must be another way using part A and B.

Any help is appreciated. Thanks.

2. ## Re: Volume and Surface area derivative problem

$\displaystyle V = \frac{1}{6\sqrt{\pi}} \cdot A^{\frac{3}{2}}$

$\displaystyle \frac{dV}{dA} = \frac{1}{4\sqrt{\pi}} \cdot A^{\frac{1}{2}}$

when $\displaystyle r = 4$ , $\displaystyle A = 64\pi$

$\displaystyle \frac{dV}{dA} = \frac{1}{4\sqrt{\pi}} \cdot 8\sqrt{\pi} = 2$

$\displaystyle dA = \frac{dV}{2} = \frac{\frac{1}{2}}{2} = \frac{1}{4} \, in^2$