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since x is decreasing at 12 knots and y is increasing at 10 knots. At t = 2, we have (since distance = speed time) y = 10 2 = 20 72 x = 12 2 = 24 x = 72 24 = 48 and, by (1), u= Substitution in (2) yields 52 du = 12(48) + 10(20) = 576 + 200 = 376 dt du 376 = 7.23 dt 52 (48)2 + (20)2 = 2304 + 400 = 2704 = 52
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which shows that, after 2 hours, the distance between ships B and C is decreasing at approximately the rate of 7.23 knots.
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20.5 The top of a 25-foot ladder, leaning against a vertical wall, is slipping down the wall at the rate of 1 foot per minute. How fast is the bottom of the ladder slipping along the ground when the bottom of the ladder is 7 feet away from the base of the wall 20.6 A cylindrical tank of radius 10 feet is being lled with wheat at the rate of 314 cubic feet per minute. How fast is the depth of the wheat increasing [Hint: The volume of a cylinder is r 2 h, where r is its radius and h its height.] 20.7 A 5-foot girl is walking toward a 20-foot lamppost at the rate of 6 feet per second. How fast is the tip of her shadow (cast by the lamp) moving 20.8 A rocket is shot vertically upward with an initial velocity of 400 feet per second. Its height s after t seconds is s = 400t 16t 2 . How fast is the distance between the rocket and an observer on the ground 1800 feet away from the launching site changing when the rocket is still rising and is 2400 feet above the ground (see Fig. 20-6) 20.9 A small funnel in the shape of a cone is being emptied of uid at the rate of 12 cubic millimeters per second. The height of the funnel is 20 millimeters and the radius of the base is 4 millimeters. How fast is the uid level dropping when the level stands 5 millimeters above the vertex of the cone [Remember that the volume of a cone is 1 r 2 h.] 3
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20.10 A balloon is being in ated by pumping in air at the rate of 2 cubic inches per second. How fast is the diameter of the balloon increasing when the radius is one-half inch 20.11 Oil from an uncapped oil well in the ocean is radiating outward in the form of a circular lm on the surface of the water. If the radius of the circle is increasing at the rate of 2 meters per minute, how fast is the area of the oil lm growing when the radius reaches 100 meters 20.12 The length of a rectangle having a constant area of 800 square millimeters is increasing at the rate of 4 millimeters per second. (a) What is the width of the rectangle at the moment when the width is decreasing at the rate of 0.5 millimeter per second (b) How fast is the diagonal of the rectangle changing when the width is 20 millimeters 20.13 A particle moves on the hyperbola x 2 18y2 = 9 in such a way that its y-coordinate increases at a constant rate of nine units per second. How fast is its x-coordinate changing when x = 9 20.14 An object moves along the graph of y = f (x). At a certain point, the slope of the curve (that is, the slope of the tangent line to the curve) is 1 and the abscissa (x-coordinate) of the object is decreasing at the rate of three units per second. At that point, 2 how fast is the ordinate (y-coordinate) of the object changing [Hint: y = f (x), and x is a function of t. So y is a composite function of t, which may be differentiated by the chain rule.] 20.15 If the radius of a sphere is increasing at the constant rate of 3 millimeters per second, how fast is the volume changing when the surface area (4 r 2 ) is 10 square millimeters 20.16 What is the radius of an expanding circle at a moment when the rate of change of its area is numerically twice as large as the rate of change of its radius 20.17 A particle moves along the curve y = 2x 2 3x 2 + 4. At a certain moment, when x = 2, the particle s x-coordinate is increasing at the rate of 0.5 unit per second. How fast is its y-coordinate changing at that moment 20.18 A plane ying parallel to the ground at a height of 4 kilometers passes over a radar station R (see Fig. 20-7). A short time later, the radar equipment reveals that the plane is 5 kilometers away and that the distance between the plane and the station is increasing at a rate of 300 kilometers per hour. At that moment, how fast is the plane moving horizontally 20.19 A boat passes a xed buoy at 9 a.m., heading due west at 3 miles per hour. Another boat passes the same buoy at 10 a.m., heading due north at 5 miles per hour. How fast is the distance between the boats changing at 11:30 a.m. 20.20 Water is pouring into an inverted cone at the rate of 3.14 cubic meters per minute. The height of the cone is 10 meters and the radius of its base is 5 meters. How fast is the water level rising when the water stands 7.5 meters in the cone
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