Related Rates of Change

Related rates problems involve finding the rate of change of one quantity in terms of the rate of change of another, using the chain rule to link them. These problems appear frequently in Further Mathematics exams and require careful setup.

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The Chain Rule for Related Rates

If $y$y depends on $x$x, and $x$x depends on $t$t (time), then:

$\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$dtdy​=dxdy​⋅dtdx​

More generally, for three or more linked variables:

$\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}$dtdA​=drdA​⋅dtdr​

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Strategy for Solving Related Rates Problems

1. Identify all the quantities and which rates are given/required.
2. Find a relationship (formula) linking the quantities.
3. Differentiate the relationship with respect to time $t$t.
4. Substitute the known values and solve for the unknown rate.

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Worked Examples

Example 1: A circle has a radius that is increasing at a rate of 2 cm/s. Find the rate of increase of the area when the radius is 5 cm.

Given: $\frac{dr}{dt} = 2$dtdr​=2. Find: $\frac{dA}{dt}$dtdA​ when $r = 5$r=5.

$A = \pi r^2$A=πr2, so $\frac{dA}{dr} = 2\pi r$drdA​=2πr.

$\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt} = 2\pi(5)(2) = 20\pi \text{ cm}^2\text{/s}$dtdA​=drdA​⋅dtdr​=2π(5)(2)=20π cm2/s