Question

man is climbing a vertical ladder at 0.2 metres/sec. there is a searchlight on the ground 15 metres from the base of the ladder, whose light beam is angled up and pointed toward the man as he climbs the wall. how fast is the searchlight beam turning (in radians per sec) when the man is 8 metres up the ladder?

Answer 1

The rate at which the searchlight beam is turning when the man is 8 meters up the ladder is approximately 0.026 radians per second.

Step-by-step explanation:

To find the rate at which the searchlight beam is turning, we need to use similar triangles. Let's denote the distance between the man and the base of the ladder as x, and the height of the man on the ladder as h.

Since the man is climbing the ladder at a constant rate of 0.2 m/s, the rate at which he is moving horizontally, dx/dt, is also 0.2 m/s.

Using the concept of similar triangles, we can set up the following equation:

(15 - x) / h = dx / dt

Substituting the given values, when the man is 8 meters up the ladder (h = 8), we can solve for the rate of change of the searchlight beam:

(15 - x) / 8 = 0.2

Simplifying, we can find that x = 11.6 meters. Substituting this value back into the equation, we find that the rate of change of the searchlight beam is approximately 0.026 radians per second.