Calculus Homework 31

  1. The radius of a sphere is increasing at a rate of 2 inches per minute. What is the rate of change of the volume of the sphere at the point in time when the radius is 6 inches?
  2. All the edges of a cube are expanding at the rate of 3 centimeters per second. How fast is the volume changing when each edge is 1 centimeter? 10 centimeters?
  3. At a sand and gravel plant, sand is falling off a conveyor onto a conical pile at the rate of 10 cubic feet per minute. The diameter of the base of the cone remains three times the altitude. At what rate is the height of the pile changing when it is 15 feet high?
  4. A trough is 12 feet long and 3 feet across the top. Its ends are isosceles triangles with an altitude of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast is the water level rising when it is 1 foot deep?
  5. A boat is pulled in by means of a winch on the dock 12 feet above the deck of the boat. The winch pulls in rope at the rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock?
  6. An air traffic controller spots two planes at the same altitude converging on a point as the fly at right angles to each other. One plane is 150 miles from the point and is moving at 450 miles per hour. The other plane is 200 miles from the point and has a speed of 600 miles per hour.
    1. At what rate is the distance between the planes decreasing?
    2. How much time does the traffic controller have to get one of the planes on a different flight path?
  7. An airplane is flying at an altitude of 6 miles and passes directly over a radar antenna. When the plane is 10 miles away, the radar detects that it is moving away at a rate of 240 miles per hour. What is the speed of the plane?
  8. A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light
    1. at what rate is the tip of his shadow moving?
    2. at what rate is the length of his shadow changing?
  9. A balloon rises at the rate of 10 feet per second from a point on the ground 100 feet from an observer. What is the rate of change of the angle of elevation of the balloon from the observer when the balloon is 100 feet above the ground?
  10. A patrol car is parked 50 feet from a long warehouse. The revolving light on top of the car rotates at a rate of 30 revolutions per minute. How fast is the light beam moving along the wall when the beam makes an angle of 30 with the line, perpendicular to the wall, from the light to the wall? How fast the light beam moving when it makes an angle of 60 ?
  11. When a certain polyatomic gas undergoes adiabatic expansion, its pressure p and volume v satisfy the equation pv1.3=k, where k is a constant. Find the relationship between the rates pand v.