Answer:
Explanation:
The rotating light is located 19 feet from the wall, and the angle between the light and the wall is changing at a rate of 10 degrees per 3 seconds. We want to find the rate at which the light's projection is moving along the wall.
To do this, we can use the formula for angular velocity:
ω = Δθ / Δt
where ω is the angular velocity, Δθ is the change in angle, and Δt is the change in time.
We can also use the fact that the tangent of the angle between the light and the wall gives us the ratio of the distance from the light to the wall to the distance from the light to the projection on the wall. In other words:
tan(θ) = x / 19
where θ is the angle between the light and the wall, and x is the distance between the light's projection on the wall and the point directly below the light on the ground.
We can take the derivative of both sides of this equation with respect to time to find the rate at which the light's projection is moving along the wall:
sec²(θ) dθ/dt = dx/dt / 19
We can rearrange this equation to solve for dx/dt:
dx/dt = 19 sec²(θ) dθ/dt
Plugging in the given values, we get:
dx/dt = 19 sec²(10°) (10° / 3 seconds)
Using a calculator, we can evaluate sec²(10°) to be approximately 1.017, and simplify the expression:
dx/dt = 19 x 1.017 x (10° / 3 seconds)
dx/dt ≈ 6.4 feet per second
Therefore, the rate at which the light's projection is moving along the wall when the angle is 10 degrees from perpendicular to the wall is approximately 6.4 feet per second.