A lamp post stands 15 m tall. A boat of height 10 m moves away from the lamp post at a speed of 13 m/min.
Let x = distance of boat from lamp post
Let s = length of the boat's shadow
Given: dtdx=13 m/min
The lamp casts light over the boat, creating two similar triangles:
- Large triangle: from lamp post top to shadow tip (height 15 m, base x+s)
- Small triangle: from boat top to shadow tip (height 10 m, base s)
Using similar triangles:
x+s15=s10
15s=10(x+s)
15s=10x+10s
5s=10x
s=2x
Differentiating both sides with respect to time:
dtds=2⋅dtdx
dtds=2×13
dtds=26 m/min
The shadow increases at a rate of 26 m/min.