210 Chapter 3 / Topics in Differentiation 22. For the camera and rocket shown in Figure 3.4.5, at what constant rate of 5 ft/ min, at what rate is sand pouring from rate is the rocket rising when the elevation angle is 7/4 the chute when the pile is 10 ft high? radians and increasing at a rate of 0.2 rad/s? 28. Wheat is poured through a chute at the rate of 10 ft / min 23. A satellite is in an elliptical orbit around the Earth. Its and falls in a conical pile whose bottom radius is always half distance r (in miles) from the center of the Earth is given by the altitude. How fast will the circumference of the base be 4995 increasing when the pile is 8 ft high? 1 + 0.12 cos e 29. An aircraft is climbing at a 30 angle to the horizontal. How where 0 is the angle measured from the point on the orbit fast is the aircraft gaining altitude if its speed is 500 mi/h? nearest the Earth's surface (see the accompanying figure). (a) Find the altitude of the satellite at perigee (the point 30. A boat is pulled into a dock by means of a rope attached to nearest the surface of the Earth) and at apogee (the point a pulley on the dock (see the accompanying figure). The farthest from the surface of the Earth). Use 3960 mi as rope is attached to the bow of the boat at a point 10 ft below the radius of the Earth. the pulley. If the rope is pulled through the pulley at a rate (b) At the instant when 9 is 120, the angle O is increasing of 20 ft/min, at what rate will the boat be approaching the at the rate of 2.7 / min. Find the altitude of the satel- dock when 125 ft of rope is out? lite and the rate at which the altitude is changing at this Pulley instant. Express the rate in units of mi/min. Boat Dock Figure Ex-30 31. For the bout in Exercise 30, how fast must the rope be pulled Apogee Perigee if we want the boat to approach the dock at a rate of 12 ft/ min at the instant when 125 ft of rope is out? 32. A man 6 ft tall is walking at the rate of 3 ft/s toward a Figure Ex-23 streetlight 18 ft high (see the accompanying figure). 24. An aircraft is flying horizontally at a constant height of (a) At what rate is his shadow length changing? 4000 ft above a fixed observation point (see the accom- (b) How fast is the tip of his shadow moving? panying figure). At a certain instant the angle of eleva- tion 0 is 30" and decreasing, and the speed of the aircraft is 300 mi/h. (a) How fast is 9 decreasing at this instant? Express the result in units of deg/s. (b) How fast is the distance between the aircraft and the observation point changing at this instant? Express the result in units of ft/s. Use I mi = 5280 fu Figure Ex-32 33. A beacon that makes one revolution every 10 s is located on a ship anchored 4 kilometers from a straight shoreline. 4000 It How fast is the beam moving along the shoreline when it makes an angle of 45 with the shore? 34. An aircraft is flying at a constant altitude with a constant Figure Ex-24 speed of 600 mi/h. An antiaircraft missile is fired on a 25. A conical water tank with vertex down has a radius of straight line perpendicular to the flight path of the aircraft 10 fi at the top and is 24 fi high. If water flows into the so that it will hit the aircraft at a point P (see the accom- tank at a rate of 20 ft /min, how fast is the depth of the panying figure). At the instant the aircraft is 2 mi from the water increasing when the water is 16 ft deep? impact point P the missile is 4 mi from P and flying at 1200 mi/h. At that instant, how rapidly is the distance between 26. Grain pouring from a chute at the rate of 8 ft / min forms a missile and aircraft decreasing? conical pile whose height is always twice its radius. How fast is the height of the pile increasing at the instant when the pile is 6 ft high? 27. Sand pouring from a chute forms a conical pile whose height is always equal to the diameter. If the height increases at a Figure Ex-34