Related rates: area and volume

About this lesson

Calculus BC (AP-aligned) › Implicit differentiation and related rates

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Related rates: area and volume

Warm-up

Three quick questions from earlier lessons.

Discover

A stone drops in a pond; the ripple's radius grows at 2 cm/s. Is the area growing at a constant rate?

  1. No, faster and faster
  2. Yes, 2 cm²/s

Key idea

When quantities are linked by an equation, their rates are linked by its derivative with respect to time.

Related rates: write an equation linking the quantities, differentiate it with respect to tt (chain rule: each quantity gets its own rate), then substitute the values at the moment of interest.

Worked example

A ripple's radius grows at 2 cm/s. How fast is the area growing when r=10r = 10 cm?

A=πr2A = \pi r^2
The link between the quantities.
dAdt=2πrdrdt\dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt}
Differentiate with respect to t.
dAdt=40π≈126\dfrac{dA}{dt} = 40\pi \approx 126 cm²/s
Now substitute r = 10, dr/dt = 2.

Your turn, guided

Your turn: a square's side grows at 3 cm/s. A=s2A = s^2. Find dAdt\dfrac{dA}{dt} when s=5s = 5.

dAdt=2sdsdt\dfrac{dA}{dt} = 2s\dfrac{ds}{dt}
Differentiate.
dAdt=30\dfrac{dA}{dt} = 30 cm²/s
2 · 5 · 3.

Practice ★

  1. A circular ripple's radius grows at 5 cm/s. How fast is its area growing when the radius is 11 cm? (cm²/s, exact)
  2. A circular ripple's radius grows at 2 cm/s. How fast is its area growing when the radius is 6 cm? (cm²/s, exact)
  1. A balloon's radius increases at 5 cm/s. With V=43πr3V = \tfrac{4}{3}\pi r^3, find dVdt\dfrac{dV}{dt} when r=3r = 3 cm. (cm³/s, exact)
  2. A balloon's radius increases at 2 cm/s. With V=43πr3V = \tfrac{4}{3}\pi r^3, find dVdt\dfrac{dV}{dt} when r=7r = 7 cm. (cm³/s, exact)

Watch out!

Practice ★★

  1. Air is pumped into a spherical balloon at 29 cm³/s. How fast is the radius increasing when r=6r = 6 cm? (cm/s, exact)
  2. Air is pumped into a spherical balloon at 26 cm³/s. How fast is the radius increasing when r=3r = 3 cm? (cm/s, exact)

Exit ticket

Three questions to check you have got it.

  1. A circular ripple's radius grows at 5 cm/s. How fast is its area growing when the radius is 11 cm? (cm²/s, exact)
  2. A circular ripple's radius grows at 2 cm/s. How fast is its area growing when the radius is 6 cm? (cm²/s, exact)
  3. A circular ripple's radius grows at 3 cm/s. How fast is its area growing when the radius is 11 cm? (cm²/s, exact)

Summary card

Related rates

  1. Equation linking the quantities.
  2. Differentiate with respect to tt.
  3. Substitute the values at that moment; solve for the unknown rate.

Watch out: substitute changing values only after differentiating.

End of the lesson

Next lesson: Related rates with right triangles Back to the chapter

Answers: Related rates: area and volume

A

  1. 110π110 \pi
  2. 24π24 \pi

B

  1. 180π180 \pi
  2. 392π392 \pi

C

  1. 29144π\frac{29}{144 \pi}
  2. 1318π\frac{13}{18 \pi}

D

  1. 110π110 \pi
  2. 24π24 \pi
  3. 66π66 \pi

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