Step by step
The question
Question: Air is pumped into a balloon at 100 cm³/s.
How fast is the radius growing when the radius is 5 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 100 cm³ per second: dV/dt = 100.
dr/dt means how fast r changes: the change each second.
We want dr/dt at the moment when r = 5.
2 Write the equation
The volume of a ball: V = 4/3 π r cubed.
Careful!
Do not put r = 5 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 = 100 and r = 5.
Multiply the numbers: 4 × 25 = 100.
5 Solve for the rate
To remove times 100π, divide both sides by 100π.
That is 1 over π.
With π ≈ 3.14159, about 0.32 cm per second.
6 Check
Let's check: a tiny moment later, 0.001 seconds.
The volume is 0.1 bigger.
Then r is about 5.000318.
r grew by 0.000318.
That is about 0.32 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.