Step by step
The question
Question: Air is pumped into a balloon at 72 cm³/s.
How fast is the radius growing when the radius is 3 cm?
1 Draw and name
Draw it.
As air goes in, the volume V grows, and so does the radius r.
The volume grows by 72 cm³ per second: dV/dt = 72.
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 volume of a ball: V = 4/3 π r cubed.
Careful!
Do not put r = 3 in yet!
r changes with time.
Numbers go in only after the derivative.
3 Take the derivative
Both V and r change with time.
The chain rule: r cubed gives 3r squared times dr/dt.
π waits.
4/3 times 3 is 4.
4 Put the numbers in
Put in dV/dt = 72 and r = 3.
Multiply the numbers: 4 × 9 = 36.
5 Solve for the rate
To remove times 36π, divide both sides by 36π.
Simplify: 2 over π.
With π ≈ 3.14159, about 0.64 cm per second.
6 Check
Let's check: a tiny moment later, 0.001 seconds.
The volume is 0.072 bigger.
Then r is about 3.000636.
r grew by 0.000636.
That is about 0.64 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.