Physics Word Problem Help?

Question

A man 6ft tall walks at a rate of 5ft/s away from a light that is 15ft above the ground. When he is 10ft away from the base of the light,

a.) At what rate is the tip of his shadow moving?
b.) At what rate is the length of his shadow changing?

Thanks SO much.
Any Help would be great.
Thanks

Answer 1

Here's a general solution first.

The man is at position xm at any instant in time
at an instant later, his position is xm+5*dt

The position of the shadow from the man can be found using trig
tan(th)=6/L where L= the horizontal distance from the man to the tip of his shadow
also, tan(th)=9/xm for any xm as long as the ground is horizontal

therefore
L=6*xm/9
and the distance from the light pole to the shadow tip, call it S is
xm+L
or
S=15*xm/9

at the position xm+5*dt
L'=(6*xm+30*dt)/9
and
S'=xm+5*dt+(6*xm+30*dt)/9
or
(15*xm+75*dt)/9

a) The speed of the shadow tip is
v=(S'-S)/dt
so
v=((15*xm+75*dt)/9-15*xm/9)/dt
or
v=75/9 ft/second
which is a constant

b) the rate of change of L is (L'-L)/dt
((6*xm+30*dt)/9-6*xm/9)/dt
or 30/9 ft/second
also a constant

this is also the speed of the shadow tip relative to the man.

j