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Question: If a man of height \(6\)ft walks at a uniform speed of \(9\)ft/sec from a lamp of height \(15\)ft th...

If a man of height 66ft walks at a uniform speed of 99ft/sec from a lamp of height 1515ft then the length of his shadow is increasing at the rate of
(A)66ft/sec
(B)1212ft/sec
(C)1010ft/sec
(D)1515ft/sec

Explanation

Solution

First of all, we have to find the length of shadow (sayxx) and then apply following formula to find the rate of increase of length of shadow, i.e., speed of shadow (vs{v_s}):
vs=dxdt{v_s} = \dfrac{{dx}}{{dt}}

Complete step-by-step answer:
Given, height of a man= 66ft
Speed of man= 99ft/sec
Height of lamp= 1515ft
We have to calculate the rate of increase of length of shadow, i.e., speed of shadow (vs{v_s}).
After ‘tt’sec, the man would move a distance of 9t9t ft away from the lamp. Let the shadow move a distance of xxft in ‘tt’sec.

Consider ΔAEC\Delta AEC and ΔBED\Delta BED;
AEC=BED=θ\angle AEC = \angle BED = \theta (common angle)
EAC=EBD=90\angle EAC = \angle EBD = {90^ \circ }
\therefore ΔAECΔBED\Delta AEC \sim \Delta BED (By AA criteria of similarity of triangles)
We know that if two triangles are similar then their corresponding sides are in the same ratio.
ACBD=AEBE\therefore \dfrac{{AC}}{{BD}} = \dfrac{{AE}}{{BE}}
On substituting the values, we get-
156=9t+xx\dfrac{{15}}{6} = \dfrac{{9t + x}}{x}
On simplifying it, we get-
15x=54t+6x\Rightarrow 15x = 54t + 6x
15x6x=54t\Rightarrow 15x - 6x = 54t
9x=54t\Rightarrow 9x = 54t
x=54t9\Rightarrow x = \dfrac{{54t}}{9}
x=6t\Rightarrow x = 6t ….. (1)
As we know that the derivative of displacement(x)\left( x \right) with respect to time (t)\left( t \right) gives us the velocity.
Therefore, velocity of shadow, vs=dxdt{v_s} = \dfrac{{dx}}{{dt}}
vs=ddt(6t)\Rightarrow {v_s} = \dfrac{d}{{dt}}\left( {6t} \right)
vs=6\Rightarrow {v_s} = 6 ft/sec
Therefore, the length of shadow is increasing at the rate of 66ft/sec.
So, option (A) is the correct answer.

Note: Here we use AA (Angle-Angle) criteria of similarity which states that if two corresponding angles of two triangles are equal, then two triangles are similar. Also, similar triangles have a property that their corresponding sides are in the same ratio.