Example 4: Revolution Speed

A light beam on a 100 ft tower rotates in a vertical circle at the rate of one revolution per second. Find the speed of the spot of light moving along the ground at a point 1000 ft from the base of the tower.

We start by drawing the picture in Figure 7.2.6.

Figure 7.2.6

Assume the rotation is counterclockwise. Let t be time and let x and θ be as in the figure.

Then

= 2π radians/sec, x = 100 tan θ ft.

We wish to find dx/dt when x = 1000.

When x = 1000,

sec² θ = 1 + tan² θ = 1 + (x/100)² = 1 + 10² = 101.

Therefore