A man with a height of $2$ m is initially positioned under a $10$ m tall lamp post and starts to

Question

A man with a height of $2$ m is initially positioned under a $10$ m tall lamp post and starts to walk away from the lamp post at a constant speed of $2$ m/s. The light ray from the top of the lamp post to the tip of the man’s shadow makes an angle of $\theta$ radians with the ground. The distance between the man and the lamp post is $x$ m as shown in the diagram below. Express $\tan \theta $ in the form $\frac{k}{x}$, where $k$ is a constant to be determined. Show that $\frac{\mathrm{d}\theta }{\mathrm{d}x}=-\frac{8}{64+{{x}^{2}}}$.[2] When the man has walked for $4$ seconds, find the rate at which $\theta$ is changing.[4]

Accepted Answer

(i) $k=8$, $\tan\theta=\dfrac8x$, and $\dfrac{\mathrm d\theta}{\mathrm dx}=-\dfrac8{64+x^2}$ (ii) $\dfrac{\mathrm d\theta}{\mathrm dt}=-\dfrac18\text{ rad s}^{-1}$

2012 JJC Promo Q8

2012 JJC Promo Q8

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