 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/9/b/solution

Differentiate before substituting the instantaneous side length. The related rates equation is

2 d \dot{d} = 2 b \dot{b}, \dot{d} = \frac{b}{a ^{2} + b ^{2}} \dot{b} .

At the stated dimensions and rate,

\dot{d} = \frac{30}{1 0 ^{2} + 3 0 ^{2}} 4 = \frac{12}{10} cm s^{- 1} \approx 3.795 cm s^{- 1} .

The diagonal grows more slowly than the changing side, since

b / d < 1

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