EXAMPLE 5 A man walks along a straight path at a speed of 3 ft/s. A searchlight is located on the ground 4 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 3 ft from the point on the path closest to the searchlight? SOLUTION We draw the figure to the left and let x be the distance from the man to the point on the path closest to the searchlight. We let 0 be the angle between the beam of the searchlight and the perpendicular to the path. We are given that dx/dt = 3 ft/s and are asked to find do/dt when x = 3. The equation that relates x and 0 can be written from the figure: X = tan(0) x = 5 x tan ( 0 ) . X Differentiating each side with respect to t, we get Video Example () LO dx = dt cos (0) X SO de d+ X m X 11 $ cos-(0) X When x = 3, the length of the beam is 5, so cos(0) = = and do = 3/5 X 5/13 X -= 15/1 X . The searchlight is rotating at a rate of /169 rad/s