A police cruiser hunting for a suspect pulls over and stops at a point 10 ft from a straight wall.

The flasher on top of the cruiser revolves at a constant rate of 75 /second, and the light beam casts a spot of light as it strikes the wall.
How fast is the spot of light moving along the wall at a point 50 ft from the point on the wall closest to the cruiser?

<---10--->

If x is the distance from the closest point on the wall, then

x = 10 tanθ
dx/dt = 10 sec^2θ dθ/dt

when x=50, tanθ = 5, so secθ = √26

dx/dt = 10(26) (75)(2π) = 39000π ft/sec

To solve this problem, we can use the concept of related rates. Since the flasher on top of the cruiser revolves at a constant rate of 75°/second, we can relate the angle of rotation to the position of the spot of light on the wall.

Let's assume that the angle formed between the light beam and the line perpendicular to the wall at the point closest to the cruiser is θ. This angle is changing as the flasher revolves.

The distance between the cruiser and the wall is given as 10 ft, and the distance between the point on the wall closest to the cruiser and the point we are interested in is 50 ft.

We want to find the rate of change, or velocity, of the spot of light along the wall at the point 50 ft from the closest point.

Now, we can set up a right triangle to represent the situation:
_____
| /
| /
| / θ
|/_____

Using this triangle, we can relate θ to the distances on the right triangle. We have a right triangle with a hypotenuse of 10 ft and an adjacent side of 50 ft. Therefore, we can use the cosine function:

cos(θ) = adjacent/hypotenuse
cos(θ) = 50/10
cos(θ) = 5

Now, we can differentiate both sides of the equation with respect to time (t) to get the rate of change:

d(cos(θ))/dt = d(5)/dt

Using the chain rule on the left side:

- sin(θ) * dθ/dt = 0

We can simplify the right side, as it is a constant:

- sin(θ) * dθ/dt = 0

Since we already have the value of cos(θ) = 5, we can find sin(θ) by using the Pythagorean identity:

sin^2(θ) + cos^2(θ) = 1
sin^2(θ) + 5^2 = 1
sin^2(θ) = 1 - 25
sin^2(θ) = -24

Since the Pythagorean identity states that sin^2(θ) cannot be negative, we have no real solution for sin(θ). Therefore, the rate of change, or velocity, of the spot of light along the wall at the point 50 ft from the closest point is 0 ft/s.

Thus, the spot of light is not moving along the wall at the point 50 ft from the closest point.