A man walks along a straight path at a speed of 3 ft/s. A searchlight is located on the ground 4 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 3 ft from the point on the path closest to the searchlight

0.48 rad/sec

Step-by-step explanation:

From the diagram:

We can find 'x' using trigonometry :

Tanθ = opposite / Adjacent

Opposite = x ; adjacent = 4

Tanθ = x / 4

x = 4tanθ

Let u = 4 and v = tanθ

If dx/dt = 3ft/s ;

dθ/dt when x = 3ft

Differentiate x with respect to t

dx/dt (4tanθ)

Let d/dθ tanθ = sec^2θ

Sec^2θ = 1 / cos^2θ

dθ/dt = 1/4cos^2θdx/dt

dθ/dt = 1/4cos^2θ(3)

dθ/dt = 3/4cos^2θ

When x = 3ft

Cosθ = Adjacent / Hypotenus

Hypotenus = √(4^2 + 3^2

Hypotenuse = √16 + 9 = √25 = 5

Cosθ = 4/5

dθ/dt = 3/4(4/5)^2

dθ/dt = 3/4(16/25)

dθ/dt = 48/100 = 12/25 = 0.48 rad/sec