Related Rates

Learn related rates for area, related rates for volume.

Lesson 2 of 32 parts Practice Flashcards

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Related Rates for Area

Objective: Explain related rates for area, apply the stated method, and check its conditions.

Related-rate problems connect time-dependent quantities with an equation, then differentiate with respect to time. For a circle, A=πr² gives dA/dt=2πr·dr/dt. Substitute the specified instant only after differentiating the relationship. Use the chain-rule factor dr/dt.

Formulas

$dA/dt=2πr·dr/dt$

Worked examples

Try itA circle has radius 7 cm increasing at 6 cm/s. Find dA/dt as a coefficient of π.Show answer

Step 1: Differentiate A=πr² with respect to t. Step 2: dA/dt=2πr·dr/dt=2π(7)(6). Step 3: The rate is 84π cm²/s; the coefficient is 84.

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Related Rates for Area

Objective: Explain related rates for area, apply the stated method, and check its conditions.

Formulas

$dA/dt=2πr·dr/dt$

Worked examples

Try itA circle has radius 7 cm increasing at 6 cm/s. Find dA/dt as a coefficient of π.Show answer

Step 1: Differentiate A=πr² with respect to t. Step 2: dA/dt=2πr·dr/dt=2π(7)(6). Step 3: The rate is 84π cm²/s; the coefficient is 84.

Related Rates for Volume

Objective: Explain related rates for volume, apply the stated method, and check its conditions.

For a sphere of time-dependent radius, V=(4/3)πr³ implies dV/dt=4πr²·dr/dt. Volume-rate units are cubic length per time. A numerical radius is an instantaneous measurement, not a constant to substitute before differentiating. Differentiate before substituting the instantaneous radius.

Formulas

$dV/dt=4πr²·dr/dt$

Worked examples

Try itA sphere has radius 7 cm increasing at 6 cm/s. Find dV/dt as a coefficient of π.Show answer

Step 1: Differentiate V=(4/3)πr³. Step 2: dV/dt=4πr²·dr/dt. Step 3: The rate is 1176π cm³/s.

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