Hi Jay.
Let's take this question step by step: first, the "snowball melts at a rate..." is clearly referring to the change in volume of the snowball with respect to time. If V = volume,
A = surface area and R = radius, then dV/dt is what this phrase means.
Now "...proportional to its surface area." means that the dV/dt (the first part of the sentence) varies directly with the surface area A. Varies directly just means that the two terms are equal, but there is a constant of proportionality (usually called k) as a factor (usually attached to the second term). That means:
dV/dt = kA.
So that's all the first sentence says.
The question is "show that the radius shrinks at a constant rate". That means the change in radius with respect to time is constant (let's call this constant q). So that means you are asked to show dr/dt is some constant.
The way to do this is to know the formulas for the volume and surface area of a sphere:
V = (4/3) π r^{3} and A = 4 π r^{2}
and take derivatives.
dV/dt is the derivative of volume with respect to time, but remember that r is also changing, so you have to use the chain rule:
dV/dt = d/dt((4/3) π r^{3}) = (4/3) π 3 r^{2 }(dr/dt)
Now let's plug that into the first equation:
(4/3) π 3 r^{2 }(dr/dt) = k A = k(4 π r^{2}) = 4 π k r^{2}
So when we simplify by dividing left and right sides by 4 π r^{2}, we get:
dr/dt = k
which is saying exactly what we wanted to prove: that dr/dt is constant.
For the second (numerical) part of the question, you don't know the
original volume (call it V again), but hopefully it will cancel out
later, so just start with what you know. Call the original radius R,
the radius after 20 minutes r, and the volume after 20 minutes v,.
V = 4/3 π R^{3}
v = 4/3 π r^{3}
v = (8/27)V
Now use this last equation to tie together the two volume equations:
4/3 π r^{3} = (8/27) (4/3 π R^{3})
so when you simplify and take the cube root,
r = 2/3 R
That's very useful, because it is saying that the new radius is 2/3 the initial radius. But since the change in radius with respect to time is constant, the radius is shrinking at a constant rate. So if the radius lost 1/3 its length in 20 minutes, it will take another 40 minutes to melt away completely.
Stephen La Rocque.>
