Sphere Differentiation

.diToday's question deals with differentiation and spheres. The question uses some very nice properties of differentiation and the dy/dx notation. It has two parts, but so do most questions from text books (apparently, people at Yahoo Answers aren't that original in making up questions).

A sphere has radius of length r cm, surface area of S cm², and a volume of V cm³ on a given instant, t.

1) Prove: (dV/dt) * (dr/dt) = (dS/dt)² / 16π

2) Find the surface area when dV/dt = π cm³/sec and dS/dt = 2π cm²/sec

Not that bad, is it?

First, there are two important formulas we are going to use in order to solve this problem:
V = 4πr³ / 3
S = 4πr²

Now differentiate each function according to t:
dV/dt = 4πr² dr/dt
dS/dt = 8πr dr/dt

To get dV/dt * dr/dt, multiply the first equation by dr/dt:
dV/dt * dr/dt = 4πr² (dr/dt)²

Now square the second equation:
(dS/dt)² = 64π²r² (dr/dt)²

Divide the second equation by 16π:
64π²r² / 16π = 4πr² (dr/dt)²

Now this is exactly the same as dV/dt * dr/dt.
Q.E.D

The second question:
dS/dt = dS/dV * dV/dt
dS/dV = dV/dt * dS/dt
dS/dV = (8πr dr/dt) / (4πr² dr/dt)
dS/dV = 2/r
dS/dt = 2/r * dV/dt
dS/dt = 2/r * π
dS/dt = 2π
2π = 2π/r
1 = 1/r
r = 1

Plug r = 1 into the surface area function:
S = 4π * 1² = 4π cm²

Problem solved.

Hope you liked it.

Nadav

nadavs