# calculus

A person 150cm tall is walking away from a lamp post at the rate of 15 meter per minute. when the man is 2.5m from the lamp post, his shadow is 3m long. Find the rate at which the length of the shadow is increasing when he is 7m from the lamp post.

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1. when the man is x from the pole, and his shadow is s,

150/300 = h/(250+300)
so, the pole's height is h=275 cm

so, at any time,

150/s = 275/(x+s)
or, more easily manipulated,

(x+s)/275 = s/150
x/275 = s/330
x = 5/6 s

so, at any time
dx/dt = 5/6 ds/dt

since dx/dt = 15,
ds/dt = 18 m/s regardless of the distance from the pole.

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