A girl of height 120 cm is walking towards a light on the ground at a speed of 0.6 m/s. Her shadow is being cast on a wall behind her that is 5 m from the light. How is the size of her shadow changing when she is 1.5 m from the light?

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Answer 1 · 2015-04-26T01:03:04

Let the Height of the girl's shadow on the wall be $h$ $h$
and the distance the girl is from the light be $w$ $w$

We are asked for $\frac{d h}{d t} where t is a measure of time$ $(dh)/(dt) " where " t " is a measure of time"$

We are given $\frac{d w}{d t} = - 60 cm/sec$ $(dw)/(dt) = -60 " cm/sec"$

and using the diagram below we can determine $\frac{d h}{d w}$ $(dh)/(dw)$
enter image source here

$h = \frac{6 \times 10^{4}}{w}$ $h = (6xx10^4)/w$

$\to \frac{d h}{d w} = - \frac{6 x 10^{4}}{w^{2}}$ $rarr (dh)/(dw) = - (6x10^4)/(w^2)$

$\frac{d h}{d t} = \frac{d h}{d w} \cdot \frac{d w}{d t}$ $(dh)/(dt) = (dh)/(dw)*(dw)/(dt)$

$= (\frac{- 6 \times 10^{4}}{w^{2}}) \cdot (- 6 x 10)$ $= ((-6xx10^4)/(w^2)) * (-6x10)$

$= \frac{36 \times 10^{5}}{w^{2}} (cm/sec)$ $=(36xx10^5)/w^2 " (cm/sec)"$

When $w = 150$ $w=150$

$\frac{d h}{d t} = \frac{36 \times 10^{5}}{150^{2}} (cm/sec)$ $(dh)/(dt) = (36xx10^5)/150^2 " (cm/sec)"$

$= \frac{36 \times 10^{3}}{225} (c \frac{m}{sec})$ $= (36xx10^3)/(225) " (cm/sec)$

$= 160 cm/sec$ $= 160 " cm/sec"$