Final answer:
The searchlight is rotating at a rate of 4ft/sec when the man is 15ft from the point on the path closest to the searchlight.
Step-by-step explanation:
To find the rate at which the searchlight is rotating when the man is 15ft from the point on the path closest to the searchlight, we can use related rates of change.
Let's denote the distance between the man and the searchlight as x. Since the man is walking straight, we have a right triangle formed by the man, the searchlight, and the point on the path closest to the searchlight. The hypotenuse of this triangle is 20ft and the side adjacent to the angle we are interested in is x.
Using Pythagorean theorem, we can write: x^2 + 15^2 = 20^2. Simplifying this equation, we get: x^2 = 400 - 225 = 175. Taking the square root of both sides, we find that x is approximately 13.23ft.
To find the rate at which the searchlight is rotating, we need to find dx/dt, the rate at which x is changing with respect to time. Since we know that the man is walking at a speed of 4ft/sec, the rate at which x is changing is also 4ft/sec.
Therefore, when the man is 15ft from the point on the path closest to the searchlight, the searchlight is rotating at a rate of 4ft/sec.
Complete Question:
A mans walks along a straight path at a speed of 4ft/sec. A scarch light is located on the ground 20ft from the path and is kept focused on man. At what rate is the scarch light rotating when the man is 15ft from the point on the path closest to the search light.