need help for calculus related rate map?

Question

If a snowball melts so that its surface area decreases at a rate of 1 cm^2/min, find the rate at which the diameter decreases when the diameter is 10cm

Answer 1

d = diameter
Surface area (SA)= pi d^2 for a sphere.
d(SA)/dt = d(SA)/dd * dd/dt Related rate eqtn
d(SA)/dd = 2 pi d
dd/dt= ?
1 cm^2/min = 63 pi * dd/dt
dd/dt = appx 0.016 cm/min

Answer 2

First, you'll need to make an assumption about the shape of the snowball. The best shape to use is obviously a "ball" shape or a sphere.
Look at what you know...They give you the rate of change of surface area, the diameter of the ball at a certain time, and you're looking for the rate of change of the diameter. So they give you dA/dd (rate of change of [surface] area with respect to diameter), and d (which is just the radius, r, doubled), and want you to find dd/dt (the change in diameter with respect to time).

The goal is to find a way to relate all of these in order to solve for dd/dt.
First, look at the surface area of a sphere.
It's
A = 4pi(r^2)
However, this uses the radius of the sphere and you need to make this equation in terms of the diameter...
You know that: diameter = 2 x radius, so it follows that...
r = d/2
Plug this in for r.
A = 4pi(d/2)^2 = 4pi(1/4)(d^2) = pi(d^2)
Now to relate the rates, differentiate implicitly. The reason you're doing this is because you're finding a rate of change with respect to time, a variable which doesn't actually show up in your equation, but you have to treat diameter of the melting snowball as dependent on/a function of time. Think of A as A(d) and d as really d(t)
So differentiate implicitly:
A = pi(d^2)
dA/dd = 2pi(d)(dd/dt)
Now all you have to do is plug in all your given values. dA/dd, the rate of change of the surface area was 1 cm^2/min. d (the diameter) was 10cm...
1 = 2pi(10)(dd/dt)
Solving for dd/dt (the rate of change of the diameter.) gives...
dd/dt = 1/20pi cm/min

Just remember that this change is negative since the snowball is melting and getting smaller. ^_^