Related Rates of Change

  1. A square, initially having sides of length 1 mile, begins growing. The length of its sides begin increasing at a constant rate of 7 mph. At what rate is the area of the square changing at the moment its area is five square miles?
  2. The length and widths of a rectangle each begin growing at a rate of 9 ft/s and 4 ft/s respectively. At the moment the length and width are 100 ft and 50 ft respectively, at what rate is the area of the rectangle changing?
  3. A cylindrical vase with a radius of two inches is being filled with water at a rate of 27 in³/hr. How fast is the water level increasing in the vase?
  4. Two ships at sea, the Albacore and the Beaumont, were resting at the same latitude though they were still 100 nautical miles apart. At noon on Monday the ships departed from rest, moving further away from each other, the Albacore heading west at a speed of 21 knots (nautical miles per hour) and the Beaumont heading north at a speed of 22 knots. How quickly was the distance between the ships increasing at 5pm Monday afternoon?
  5. A 17 ft long ladder is resting with its top against a vertical wall. Suddenly it loses traction; its top begins sliding down the wall at a constant speed of 3 ft/s, while its base begins sliding along the ground away from the wall.
    1. How fast is the base of the ladder sliding along the ground the moment the base is 8 ft from the wall?
    2. How fast is the angle between the top of the ladder and the wall changing at the moment the angle is π/3 radians?
  6. A boat is tethered to a dock by a rope fed through a crank fastened to the dock. The cleat on the bow of the boat at which the rope is tied is exactly five meters lower than the crank on the dock. If the rope is reeled in at a constant rate of 2 m/s, at what rate will the boat be approaching the dock the moment there only 23 m of rope out?
  7. You are flying a kite at a constant altitude of 115 ft. You begin rapidly letting out kite-string at a constant rate of 36 ft/s as the wind carries the kite horizontally away from you, maintaining that constant altitude. What is the instantaneous velocity of the kite the moment you’ve let out 277 ft of kite string?
  8. It’s midnight. A 5'-tall man walks on a sidewalk, under and past a streetlamp mounted at the top of a 24'-tall lamp post. If the man is walking at a pace of 6 ft/s away from the post, how fast is the tip of his shadow moving along the sidewalk at the moment he is 36' from the pole?
  9. Suppose you are blowing up a perfectly spherical balloon. Per the force and control of your lungs, mid-breath, the instant the balloon has a radius of 10 cm you are inflating it at a rate of 2000 cm³/s.
    1. How fast is the radius of the balloon increasing at this instant?
    2. How fast is the surface area of the balloon increasing at this instant?

  10. A sprinter is about to run along a straight portion of track. There is a radar gun mounted on a post 17 ft from the track that will automatically swivel to track the sprinter as she passes. The sprinter starts, maintaining a constant speed of 29 ft/s.

    1. After passing the radar gun and continuing on, how fast is the distance between the sprinter and the radar gun increasing the moment that she is 145 ft from it?
    2. How fast is the head of the radar gun swiveling (in radians-per-second) the moment that the sprinter is 145 ft from it?
  11. Imagine a water tank in the shape of an inverted cone, having a large circular opening at the top that narrows to a small spigot at the bottom. The tank is 10 m tall and the radius of the top opening is 4 m. Initially the tank is full of water, but someone opens the spigot at the bottom, draining the water at a constant rate of 5 m³/min. How fast is the water level in the tank dropping at the moment the depth of the water is 6 m?
  12. Imagine a cylindrical water tank with a height of 10 m and base-radius of 4 m. Someone opens a spigot at the base of the tank and water begins gushing from the tank at a variable rate of \(x^2\) m³/min, where \(x\) is the water level in the tank at that moment. How fast is the water level in the tank dropping at the moment the volume of water left in the tank is 240 m³?
  13. A police cruiser is pursuing a truck south towards an intersection. The truck turns left at the intersection, heading east. At the moment the police cruiser is still 560 ft north of the intersection, the office pulls out his radar gun and points it at the truck. At this exact moment the cruiser is travelling at 90 mph, the truck is already 900 ft east of the intersection, and the offer’s radar gun displays that the distance between his cruiser and the truck is increasing at a rate of 105 mph. How fast is the truck going? (Be mindful of units.)

  14. Two cars are sitting next to each other on a salt flat in the Utah desert, bumper-to-bumper, one pointing north and the other pointing east. At the same moment, they each take off in the direction they’re pointed, the first car at 104 mph and the second car at 153 mph.

    1. At what rate is the distance between the cars increasing 1 minute after they take off?
    2. Suppose that instead of taking off from the same spot, the second car got a ¼ mile head start eastward. Now at what rate is the distance between the cars increasing 1 minute after they take off?
    3. Suppose that instead of pointing in orthogonal directions, the second car were pointing south-east. Now at what rate is the distance between the cars increasing 1 minute after they take off?
  15. Eberhart A swing consists of a board at the end of a 10 ft long rope. Think of the board as a point \(P\) at the end of the rope, and let \(Q\) be the point of attachment at the other end. Suppose that the swing is directly below \(Q\) at time \(t=0\) and is being pushed by someone who walks at the speed of 6 ft/s from left to right. Find (a) the speed the swing is rising after one second and (b) the angular speed of the rope in radians-per-second after one second.
  16. Knill The ideal gas law \(pV = T\) relates pressure \(p\) and volume \(V\) and temperature \(T\). Assume the temperature \(T = 50\) is fixed and \(\dot V=-5\). Find the rate \(\dot p\) with which the pressure increases when \(V = 10\) and \(p= 5.\)
  17. Knill There are cosmological models which see our universe as a four dimensional sphere which expands in space time. Assume the volume \(V = \frac 1 2 \pi^2 r^2\) increases at a rate \(\dot V = 100\pi^2r^2\), What is \(\dot r\)? Evaluate it for \(r = 47\) (billion light years).

  18. On pleasant summer day when the wind is too lazy to blow and the sea is calm, an oil rig off the Louisiana coast begins to leak oil at a constant rate of one barrel-per-second, forming a circular layer of oil on the surface of the ocean centered at the rig. Assuming that the average thickness of this oil slick is ½ cm, how quickly must the edge of the oil slick be moving 1 minute after the leak began? How quickly must the edge of the oil slick be moving 10 minute after the leak began? What about 1 hour after the leak began? What about 1 day?

  19. Tabrizian Suppose that the minute-hand of an analogue clock is 5" long and the hour-hand is 2" long. How fast is the distance between the tips of the hour- and minute-hand changing at 2 pm?
  20. Little Cindy-Sue is about to lose her first tooth. Her dad, knowing no better, decides to employ the age-old tooth-extraction technique of tying one end of a piece of string to the tooth, the other end to a doorknob, and then slamming the door shut. He gets a 4' piece of string and ties it to a doorknob that is 3' away from the hinges of the door. Opening the door to a right-angle, flush with the wall, he places Cindy-Sue along the same wall 3.228' away from the doorknob, and ties the other end of the string to her tooth. Luckily, Cindy-Sue’s tooth is exactly as high off the ground as the doorknob. Excited now, he swings the door, slamming it from against the wall to fully closed (at a constant angular speed) in ⅛ of a second, successfully pulling the tooth! At what velocity was the distance between Cindy-Sue's tooth and the doorknob increasing at the moment the string become taught and the tooth yanked out?