related rates problems?

Question

A boy 5 ft. tall is walking away from a street at the rate of 3 ft./sec.. If the light is 12 ft. above the level ground, determine (a) the rate at which the tip of his shadow is lengthening, (b) the rate at which the tip of his shadow is moving and (c) the rate at which his head is receding from the light when he is 24 ft from the point directly below the light .

Answer 1

Imagine similar triangles or draw a diagram of a light-poll, boy and the tip of the shadow,


s1 / (h1-h2)= s2 / h2
h1- height of the light
h2 - height of the boy
s1- horizontal distance from the light poll to the boy
s2 - horizontal distance from the boy to the tip of the shadow

we can also express it as rates of change with respest to time or speeds.
ds1/dt(1/(h1-h2)= ds2/dt (1/h2)
then ds2/dt=ds1/dt(h2/(h1-h2)
ds2/dt=ds1/dt(h2/(h1-h2)
ds2/dt=5(5/(12+5))=25/17 ft/s
(b)
same principal different different triangles.
s2/h1=s1/(h1-h2)
now s2 is the distance from the tip of the shadow to the base of the lamppost.

ds2/dt=ds1/dt(h1/(h1-h2))
ds2/dt=5(12/(12-5))
ds2/dt=60/7 ft/s

c) I'll let you do this one
Hint let S3 be the distance from the light to the boy's head

S3=sqrt((h1-h2)^2 + S1^2)
S1=Vt=24 ft
treat S1 as function of time.

I hope it was useful.
Let me know if not