A man 180 cm tall walks at a rate of 2 m/sec. away, from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3 m away from the base of light?

Given: man 180cm tall walks at a rate of 2 m/sec away; from a source of light that is 9 m above the ground


To find the rate at which the length of his shadow increases when he is 3m away from the pole


Let AB be the lamp post and let MN be the man of height 180cm or 1.8m.


Let AL = l meter and MS be the shadow of the man


Let length of the shadow MS = s (as shown in the below figure)



Given man walks at the speed of 2m/sec



So the rate at which the length of the man’s shadow increases will be


Consider ΔASB




Now consider ΔMSN, we get




So from equation(ii) and (iii),






Applying derivative with respect to time on both sides we get,




[from equation(i)]



Hence the rate at which the length of his shadow increases by 0.6 m/sec, and it is independent to the current distance of the man from the base of the light.


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