How can i answer this question?

Question

A surface area of a sphere is decreasing at a constant rate of 4πcm2s -1. At what rate is the volume decreasing at the instant when its radius is 2 cm?

Answer 1

Hi Tasneem,

Let's start by looking at some relevant equations for the Volume (V) and Surface Area (A) of a sphere:

V = 4/3 * π * r³

A = 4 * π * r²

* Side note, the Area equation is a derivative of the Volume equation! dV/dr = 4 * π * r²

The problem gives us a rate of change of -4π cm²/s for the area, and is asking us for the rate of change of the volume. So, we are given the derivative of Surface Area relative to time, dA/dt. We are looking for dV/dt. We are also given the radius (r) at that moment.

Since we don't have equations for the Area or Radius in terms of time, we want to modify them to something we do have - the radius, which we are given in the problem and our useful equations! So, we'll relate the rates of change of the Area and Volume in terms of the change of the radius:

dA/dt = dA/dr * dr/dt

dV/dt = dV/dr * dr/dt

We will rearrange these for our first approach:

(dA/dt)/(dA/dr) = dr/dt

(dV/dt)/(dV/dr) = dr/dt

dA/dr is the derivative of 4 * π * r², which we find to be 8 * π * r. dV/dr is the derivative of 4/3 * π * r³, which we know is the same as A: 4 * π * r².

At this point, we can fill in the derivatives with what we know:

dA/dt = -4π cm²/s (given)

dA/dr = 8π * r, and given r = 2, dA/dr = 8*2*π, or 16π cm.

dV/dr = 4π * r², and given r = 2, dV/dr = 4*4*π, or 16π cm².

Since we know (dA/dt)/(dA/dr) = dr/dt and (dV/dt)/(dV/dr) = dr/dt, we can set the two sides of the equations equal to eachother:

(dA/dt)/(dA/dr) = (dV/dt)/(dV/dr)

Filling in, we get:

(-4π)/(16π) = (dV/dt)/(16π), which results in: dV/dt = -4π cm³/s.

So our final answer is -4π cm³/s.

Another way to conceptualize this from the halfway point:

dA/dt = dA/dr * dr/dt

dV/dt = dV/dr * dr/dt

We can find dA/dr by deriving A = 4 * π * r², which gives us dA/dr = 8π * r. Plugging in r = 2 here, we get dA/dr = 16π.

Since we know dA/dt (given) as 4π, we can solve for dr/dt:

-4π = 16π * dr/dt

and find that dr/dt = -1/4.

Then, we can plug that into our second equation:

dV/dt = dV/dr * -1/4

and we can find dV/dr by deriving V = 4/3 * π * r³, which becomes dV/dr = 4 * π * r². Plugging in for r = 2, we find dV/dr = 16π: And that leaves our equation as:

dV/dt = 16π * -1/4 = -4π.

You can follow the units through the equations to see that the final answer is -4π cm³/s.