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Question

A man 6′ tall moves away from a source of light 20′ above the ground level, his rate of walking being 4 m.p.h. At what rate is the tip of his shadow moving?

A
127
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B
37
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C
407
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D
None of these
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Solution

The correct option is C 407
Let x be the distance between the man and the street light and let s be the length of his shadow.

GIVEN: dxdt=4.

Distance of tip: s+x

Therefore rate of tip: dsdt+dxdt

Let us find dsdt when x=30.

Note that we have two similar triangles: the overall triangle with height 20 and base length s+x and the smaller one inside of it with height 6 and base length s. We want to solve for s in terms of x by using the ratio

6s=20s+x

6(s+x)=20s

6s+6x=20s

14s=6x

s=614x

s=37x

Differentiate both sides we get

dsdt=37dxdt

dsdt=127(dxdt=4).

Therefore rate of tip: dsdt+dxdt=127+4=407



751841_735488_ans_2615a2d00cb94c0c8d397ba6abd081d4.png

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