webassign.net C + 88 C A Ladder 10 Ft Long Rests Against A Ve... x Homework: 8B (Related Rates) - Math... 6 Sympatric speciation Definition and Ex... Free Grammar Check | QuillBot Al Homework Help - Q&A from Online Tut... EXAMPLE 5 A man walks along a straight path at a speed of 6 ft/s. A searchlight is located on the ground 24 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 10 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 A be the angle between the beam of the searchlight and the perpendicular to the path. X We are given that x/ at = 6 ft/s and are asked to find A/ at when x = 10. The equation that relates x and A can be written from the figure: A X = tan A 24 X = 24 tan A Video Example () Differentiating each side with respect to t, we get dx dA - = 24 sec-(A) . dt dt so dA 1 cos- ( 0 ) dx 1 cos ( 0) X X .(6) dt 24 24 dt 24 24 COS2(A) When x = 10, the length of the beam is 26, so cos(A) = 24/26 = 1/4 X and dA = 1/4 ( 24/26 )2 = 36/169 dt The searchlight is rotating at a rate of 36/169 rad/s. Need Help? Read It