Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 20) RELATED RATES 149Fig. 20-2Now we must find x, y, and U after 2 hours. Since x is increasing at the constant rate of 300 kilometers perhour and t is measured from the beginning of the flight, x = 300t (distance = speed x time, when speed isconstant). Similarly, y = 400t. Hence, at t = 2,andSubstituting in (4),x = 300(2) = 600 y = 400(2) = 800U’ = (600)2 + (800)2 = 360000 + 64OOOO = 1 OOOOOOU = 1OOOdudtdudtloo0 - 300(600) + 400(800) = 180000 + 320000 = 500000---500000 = 500 kilometers per hourSolved Problems20.1 Air is leaking out of a spherical balloon at the rate of 3 cubic inches per minute. When the radiusis 5 inches, how fast is the radius decreasing?Since air is leaking out at the rate of 3 cubic inches per minute, the volume V of the balloon isdecreasing at the rate of dV/dt = -3. But the volume of a sphere of radius r is V = 3m3. Hence,Substituting r = 5,-- dr 3---%-- % -0.00955dt l00n 314Thus, when the radius is 5 inches, the radius is decreasing at about 0.01 inch per minute.20.2 A 13-foot ladder leans against a vertical wall (see Fig. 20-3). If the bottom of the ladder isslipping away from the base of the wall at the rate of 2 feet per second, how fast is the top of theladder moving down the wall when the bottom of the ladder is 5 feet from the base?