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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 2026-10-18

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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