Step by step
The question
Question: The area of a circle grows at 30 cm²/s.
How fast is the radius growing when the radius is 3 cm?
1 Draw and name
Draw it.
The radius r grows, so the area A grows too.
The area grows by 30 cm² per second: dA/dt = 30.
dr/dt means how fast r changes: the change each second.
We want dr/dt at the moment when r = 3.
2 Write the equation
The area of a circle: A = π r squared.
Careful!
Do not put r = 3 in yet!
r changes with time.
Numbers go in only after the derivative.
3 Take the derivative
Both A and r change with time.
The chain rule: r squared gives 2r times dr/dt.
π just waits.
4 Put the numbers in
Put in dA/dt = 30 and r = 3.
Multiply the numbers: 2 × 3 = 6.
5 Solve for the rate
To remove times 6π, divide both sides by 6π.
Simplify: 5 over π.
With π ≈ 3.14159, about 1.59 cm per second.
6 Check
Let's check: a tiny moment later, 0.001 seconds.
The area is 0.03 bigger.
Then r is about 3.001591.
r grew by 0.001591.
That is about 1.59 cm per second.
Right!
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.