Question

a prisoner walks along a straight path a at a rate of 5m/s. A spot is on the ground 10m from the path and rotates to keep shinning on the prisoner. At what rate is the light rotating when the man is 20m from the point on the path closes to the light and is walking away from that point

Answer 1

Imagine a triangle with the corners at
the spotlight
the point on the path closest to the spotlight
the prisoner

One leg of the triangle (x) is growing at 1.5m/s as the prisoner walks away.
The other leg is 10m long.
The rotation of the spotlight (Θ) is the opposite angle to x.
tan(Θ) = x / 10
Θ = arctan(x / 10)

dΘ/dx = 1 / (1 + [x/10]²) * 1/10
= 1 / (10 + x²/10)

the chain rule
dΘ/dt = dΘ/dx * dx/dt
= 1 / (10 + x²/10) * 1.5

sub in 20 for x
= 1 / (10 + 400/10) * 1.5
= 1.5 / 50
= 3 / 100
So when the prisoner is at 20m, the light is rotating at 0.03 rad/s

Answer 2

hope that helps

Answer 3

thank you