Related Rates of Change

  1. Consider a spherical bubble that’s being inflated in such a way that its radius is increasing at a constant rate of ½″ per minute. At what rate is the volume of the bubble increasing the moment that the radius is 3″?
  2. Consider a spherical bubble that’s being inflated at a constant rate of 2 in³/min. At what rate is the radius of the sphere increasing the moment that the volume is 18π in³?
  3. A square, initially having sides of length 1′, begins growing. The length of its sides begin increasing at a constant rate of 7′ per hour. At what rate is the area of the square changing at the moment its side-length is 10′?
  4. A square, initially having sides of length 1′, begins growing. The length of its sides begin increasing at a constant rate of 7′ per hour. At what rate is the area of the square changing at the moment its area is five square feet?
  5. A magic cube of jello, initially 3 mm on each side, is beginning to grow before your eyes. If a side-length of the cube is growing at a constant rate of 4 mm/s, at what rate is the volume changing the moment a side is 5 mm long?
  6. A cylindrical vase with a radius of 7 cm is being filled with water at a rate of 44 cm³/hr. How fast is the water level increasing in the vase?

  7. Recall that for a triangle with a sides of lengths \(A\) and \(B\) and an angle of measure \(\theta\) between those sides, the area of the triangle is \({\tfrac{1}{2}AB\sin(\theta).}\)

    1. Suppose \(A = 5″\) and \(B = 2″\) and the angle \(\theta\) is opening (increasing) at a rate of \(\frac{1}{4}\) radian-per-second. How fast is the area changing at the moment the angle \(\theta\) is \(\tfrac{2\pi}{3}?\)
    2. Suppose \(A = 5″\) and \(B = 2″\) and the area of the triangle is increasing at a rate of 1 in²/s. At what rate is \(\theta\) changing at the moment \(\theta = \tfrac{\pi}{6}?\)
    3. Suppose \(A = 5″\) and \(B = 2″\) and the area of the triangle is increasing at a rate of 1 in²/s. At what rate is \(\theta\) changing at the moment the area is 4 in²?
    4. Suppose now that \(A = 5″\) but that \(B″\) is increasing at a rate of three in/s while \(\theta\) is increasing at a rate of \(\tfrac{1}{4}\) radian-per-second. At a certain moment in time, \(\theta = \tfrac{\pi}{6}\) and \(B = 8″.\) At what rate is the area changing at this moment?
  8. A ladybug is rapidly crawling up a vertical brick wall at TK in/s. There is a motion-sensitive swiveling spotlight anchored TK″ away from the base of the wall, that has just detected the ladybug, illuminated it, and is now tracking the ladybug up the wall. How quickly, in radians-per-second, is the spotlight swiveling upward the moment the ladybug is TK″ up the wall?
  9. A baseball diamond is 90 ft square. There are runners on first and second. The batter hits an infield ground ball which is thrown to third base for a force out. The third baseman then sees the batter running to first at TK ft/s. How far is the runner from the third baseman after he’s run 56′ towards first base? At what rate is the distance between the runner and the third baseman increasing at this same moment? From the third baseman’s perspective, at what rate, in radians-per-second, is the angle between the runner and home plate changing at this same moment?
  10. On a brisk Monday morning, two ships, the Albacore and the Beaumont, were resting at sea at the same latitude, the Albacore 100 nautical miles west of the Beaumont. At noon on Monday the ships departed from rest, 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?

  11. 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 \(\tfrac{\pi}{3}\) radians?
  12. 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 ten feet below the height of the crank on the dock. If the rope is reeled in at a constant rate of four ft/s, at what rate will the boat be approaching the dock along the water the moment there’s only 46 feet of rope out?
  13. 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 velocity of the kite at the moment you’ve let out 277 ft of kite string?
  14. 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?
  15. 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. How fast is the radius of the balloon increasing at this instant? How fast is the surface area of the balloon increasing at this instant?
  16. A sector of a circle can be folded up into a cone. Some cheap water cups are constructed this way: start with a disk of waxed paper, make two cuts to remove a small sector of the disk, and adhere the disk along the two cuts to make a “cup”. (Obviously it’s more complicated than this; you need to leave a little paper tab to glue across the cut, but let’s keep this problem simple.) Suppose the initial circular sector has radius \(r\) and has a subtended angle measuring \(\theta.\)

  17. A sprinter is about to run along a straight portion of a running 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 running, 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?
  18. Larry the lineman just secured the end of guy wire to the top of a TK-foot tall utility pole. On the ground, holding the spool of wire to his chest 5 ft off the ground, Larry walks directly away from the pole at a constant rate of 3 ft/s looking for a good spot on the ground to secure the guy wire, letting the spool unravel as he walks. At what rate is wire being let out from the spool when Larry is TK ft from the pole? At this same moment, at what rate in radians-per-second is the angle between the wire and the utility pole increasing?
  19. 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/s. How fast is the water level in the tank dropping at the moment the depth of the water is 6 m?
  20. 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/s, 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?
  21. 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 officer 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.)

  22. Two cars in the middle of the Utah desert are sitting directly next to each other, bumper to bumper, one pointing due north and the other pointing due east. At the exact same moment they each take off in the direction they’re pointed, the first at 96 mph and the second at 110 mph.

    1. At what rate is the distance between the two cars increasing ½ hour after the moment they take off?
    2. Suppose that instead of the cars taking off at the same moment, the driver of the first, north-pointing car waits until the second car has already driven 35 miles east before they take off. At what rate is the distance between the two cars increasing ½ hour after the moment the north-pointing car takes off?
    3. Suppose that instead of due east, the second car takes off in the direction due southeast of the cars’ common starting point. Now at what rate is the distance between the two cars increasing ½ hour after the moment they take off?
  23. On a 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? (Note that 1 barrel is 42 gallons, is 159 liters, is 159,000 cm³.)
  24. Eberhart A simple 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.
  25. 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 \(p = 5.\)
  26. Knill There are cosmological models which see our universe as a four dimensional sphere which expands in space time. Assume the volume \(V = \tfrac{1}{2}\pi^2 r^2\) increases at a rate \(\dot V = 100\pi^2r^2\), What is \(\dot r\)? Evaluate it for \(r = 47\) (billion light years).
  27. Tabrizian Suppose that the minute-hand of an analogue clock is 5″ long and the hour-hand is 3″ long. How fast is the distance between the tips of the hour- and minute-hand changing at 2pm?
  28. The London Eye is a 120-meter diameter Ferris wheel on the River Thames. Assume the bottom of the London Eye is at ground level. If the wheel is moving at a rate of one rotation every half hour, how quickly is a rider’s distance from ground level changing the moment that rider is 50 meters off the ground?
  29. Little Cindy Sue is about to lose her first tooth. Her dad, giddy with excitement, decides to employ the age-old tooth-extraction technique of tying one end of a 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. He opens the door so it is perpendicular to the wall and places Cindy Sue in line with the door 3.228′ away from the doorknob to allow for some slack in the string, 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. Now too excited to contain himself any long, Cindy Sue’s dad swings the door, slamming it from open to fully closed in ⅛ of a second (at a constant angular speed), successfully pulling the tooth! At what velocity was the distance between Cindy Sue’s tooth and the doorknob increasing at the moment the string became taught and the tooth was yanked out?
    30. -->