Applications of Calculus: Example 1

Question:

A spotlight on a lighthouse 140m from shore performs 1 revolution every 20 seconds. Assuming that the shoreline is straight, calculate the speed at which the spotlight is sweeping across the shoreline as it lights up the sand 300m from point B.

Answer:

1 revolution every 20 seconds is equal to 6π rads/minute.

Since the light revolves three times per minute, the angle theta changes at a rate of 3 x 2π

radians per minute; that is,

Using the triangle ABC (and the fact that it is a right triangle), we see that:

The rate at which the ray of light moves along the shoreline is:

If x = 300m then we can use Pythagoras to find the hypotenuse:

The secant is the reciprocal of the cosine. It is the ratio of the hypotenuse to the side adjacent to a given angle in a right triangle. We can use this to find sec𝜃.