Step by step
The question
Question: The radius of a circle grows at 2 cm/s.
How fast is the area growing when the radius is 5 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 2 cm per second: dr/dt = 2.
dr/dt means how fast r changes: the change each second.
We want dA/dt at the moment when r = 5.
2 Write the equation
The area of a circle: A = π r squared.
Careful!
Do not put r = 5 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 = 5 and dr/dt = 2.
5 Solve for the rate
Multiply the numbers: 2 times 5 times 2 is 20.
So it is 20π.
With π ≈ 3.14159, about 62.83 cm² per second.
6 Check
Let's check: a tiny moment later, 0.001 seconds.
r is 5.002.
The area grows by about 0.06284.
That is about 62.84 per second.
Right!
7 Pick the choice
Our answer is The area grows at 20π ≈ 62.83 cm²/s..
That is choice B.
8 Why not the others
A is 31.42.
Wrong: lost the 2: dA/dt = 2πr × dr/dt.
C is 78.54.
Wrong: that is the area, not its rate.
D is 12.57.
Wrong: used r = 1.
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.