A street light is at the top of a 16 ft pole.

Question

A street light is at the top of a 16 ft pole. A 6 ft tall girl walks along a straight path away from the pole with a speed of 6 ft/sec.

At what rate is the tip of her shadow moving away from the light (ie. away from the top of the pole) when the girl is 28 ft away from the pole?

Answer=

How fast is her shadow lengthening?

Answer=

Answer 1

say x= distance to girl

w=shadow length girl's foot to shadow tip.

then w/6=(x+w)/16,,,,,,by similar triangles

w=(3/5)x

right away

dw/dt=(3/5)dx/dt

since dx/dt=6

dw/dt=18/5 ft/s which answers second part of question

the hypotenuse squared for any x is

h²=(x+w)²+16²

differentiating wrt time

2hh'=2(x+w)(dx/dt+dw/dt)

at x=28,,,,w=84/5 and h=√(16²+(224/5)²)=47.6,,,,,224/5=(84/5)+28

dh/dt=(224/5)(6+18/5)/16

dh/dt=28.9 ft/s.

when the 6 ft. girl is 28 feet away from light atop 16 ft. pole

h is distance from atop pole to tip of shadow