Step by step
The question
Question: The radius of a circle grows at 3 cm/s.
How fast is the area growing when the radius is 10 cm?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Draw and name
Draw it.
The radius r grows, so the area A grows too.
The radius grows by 3 cm per second: dr/dt = 3.
dr/dt means how fast r changes: the change each second.
We want dA/dt at the moment when r = 10.
2 Write the equation
The area of a circle: A = π r squared.
Careful!
Do not put r = 10 in yet!
r changes with time.
Numbers go in only after the derivative.
The right way: A = πr².
3 Take the derivative
Both A and r change with time.
The chain rule: r squared gives 2r times dr/dt.
π stays.
4 Put the numbers in
Put in r = 10 and dr/dt = 3.
5 Solve for the rate
Multiply the numbers: 2 times 10 times 3 is 60.
So it is 60π.
With π ≈ 3.14159, about 188.50 cm² per second.
6 Check
Let's check: a tiny moment later, 0.001 seconds.
r is 10.003.
The area grows by about 0.18852.
That is about 188.52 per second.
Right!
7 Pick the choice
Our answer is The area grows at 60π ≈ 188.50 cm²/s..
That is choice A.
8 Why not the others
B is 30π.
Wrong: lost the 2: dA/dt = 2πr × dr/dt.
C is 100π.
Wrong: that is the area, not its rate.
D is 6π.
Wrong: used r = 1: the radius is 10.
Let's look back.
First: draw and name.
Then: write the equation.
Then: take the derivative.
Then: put the numbers in.
Then: solve for the rate.
Last: check.