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explain solution RD Sharma class 12 chapter Derivative As a Rate Measure exercise 12.2 question 8 maths

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Answer\frac{d s}{d t}=\frac{5}{2}=2.5\; km/hr

Hint: The rate at which the length of the man’s shadow increase will be  \frac{ds}{dt} .

Given:  A man 2 meters high walk at a uniform speed of 5 km/hr away from lamp post 6 meters high.


Suppose AB  the lamp post and let MN  be of the man of height 2 m.

Suppose AM = l meter and MS be the shadow of the man

Suppose length of the shadow MS=s

Given as man walk at the speed of 5 km/hr

\frac{d I}{d t}=5\; km/hr

So considering \Delta ASB,

\tan \theta=\frac{A B}{A S}=\frac{6}{I+s}................(i)

Then considering \Delta MNS,

\tan \theta=\frac{M N}{M S}=\frac{2}{s}...................(ii)

From equation (i) and (ii)

\begin{aligned} &\frac{6}{l+s}=\frac{2}{s} \\\\ &6 s=2(I+s) \\\\ &6 s=(2 I+2 s) \end{aligned}

\begin{aligned} &6 s-2 s=2 l \\\\ &4 s=2 l \\\\ &2 s=1 \end{aligned}..............(iii)

By applying derivative, With respect to time on both side

\frac{d l}{d t}=\frac{d(2 s)}{d t}                            I=2 s\left(\text { from } e q^{n} \text { iii }\right)

\begin{aligned} &\frac{d l}{d t}=2 \frac{d s}{d t} \\\\ &5=2 \frac{d s}{d t} \end{aligned}                        \left(\frac{d l}{d t}=5\right)(\text { given })

\begin{aligned} &\frac{5}{2}=\frac{d s}{d t} \\\\ &2.5=\frac{d s}{d t} \end{aligned}

Thus, the rate at which the length of his shadow increases by 2.5 km/hr

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