Related Rates: A searchlight rotates at a rate of 3

Question

A searchlight rotates at a rate of 3 revolutions per minute. The beam hits a wall located 14 miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per hour) is this dot moving when the angle  between the beam and the line through the searchlight perpendicular to the wall is 

at this point I am stuck i have gotten answers ranging from 14,442 to 26,321 to 8,423

I need help please

Answer 1

You have to find a way to relate the angle of the beam of light with how far along the wall the beam is. I suggest using graph paper. Draw the searchlight at the origin of a coordinate plane and a horizontal line at y = 14. Draw another line from the origin to another point on the horizontal line in the first quadrant. This represents the beam of light hitting the wall. Draw a vertical line from the point where the horizontal line and your new line intersect. Call theta the angle between your new line and the x axis.

Relate θ with x.

You should get, tan θ = 14 / x

Take derivatives with respect to t

sec²θ dθ/dt = -14/x² dx/dt.

x²sec²θ dθ/dt = -14 dx/dt.

x²/ cos²θ dθ/dt = -14 dx/dt.

(x/cosθ)² dθ/dt = -14 dx/dt.

x/cosθ is the hypotenuse of the triangle, so at the point when the beam is perpendicular to the wall, x/cosθ=14

Plug that in and you get

14 dθ/dt = -dx/dt

What you're solving for is how fast along the wall the dot is moving, so that's dx/dt.

Use the 3 revolutions per minute and change that into an appropriate dθ/dt and solve