Thread: Derivative word problem

I'm having a really hard time with this question:
A searchlight rotates at a rate of 3 revolutions per minute. The beam hits a wall located 8 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 ?. Note that .

I know what I need to do: relate two variables, differentiate, and solve. But I can't figure out how to relate the given variables (which might say more about my reading comprehension than calculus skills :|). Would anybody mind helping me set it up?

Notice that the beam of light, the wall, and "the line through the searchlight perpendicular to the wall" form a right triangle. Figure out an expression for the two legs of the triangle and relate it to the angle they mention.

What trig expression relates the two legs of a triangle with an angle inside the triangle?

OK, I've got tan = opposite/adjacent. Is that the right trig function to use? Most of these problems use tangent, but in this question the wording makes it seem like the given (8 miles) comes from the searchlight, which is the hypotenuse in my drawing.

I realize it's a little bit ambiguous, but my interpretation of the problem is that the distance "8" represents the shortest distance between the wall and the light tower. Thus, it would not be the hypotenuse, but instead one of the legs.

I attached a picture of how I interpret the problem.

If my interpretation is correct, then our equation should be



Does this help?

That does help! I got the answer. Thanks.