Step by step
The question
Question: The radius of a balloon grows at 1 cm/s.
How fast is the volume 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 radius grows by 1 cm per second: dr/dt = 1.
dr/dt means how fast r changes: the change each second.
We want dV/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 r = 5 and dr/dt = 1.
5 Solve for the rate
Multiply the numbers: 100π.
With π ≈ 3.14159, about 314.16 cm³ per second.
6 Check
Let's check: a tiny moment later, 0.001 seconds.
r is 5.001.
The volume grows by about 0.31422.
That is about 314.22 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.