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Related rates practice problems

Ladders, balloons, tanks and shadows: how one rate of change drives another. 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

Interactive: Related Rates Explorer

5 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 3.92

medium
Car A drives east from an intersection at 59 mph and car B drives north from it at 57 mph. How fast is the distance between them changing when A is 9 mi and B is 12 mi from the intersection?
  1. With A at distance a and B at distance b, the distance between them satisfies D² = a² + b².
    Reviewed
  2. Differentiate: 2D dD/dt = 2a da/dt + 2b db/dt.
    Reviewed
  3. \[ \left. \sqrt{x^{2} + y^{2}} \right|_{\substack{ x=9\\ y=12 }} = 15 \]
    D at that moment.✓ Proved
  4. \[ \left. \frac{59 x + 57 y}{\sqrt{x^{2} + y^{2}}} \right|_{\substack{ x=9\\ y=12 }} = 81 \]
    dD/dt = (a·da/dt + b·db/dt)/D with a = 9, b = 12.✓ Proved
Answer \( 81 \ \text{mph} \approx 81\ \text{mph} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.93

medium
A 25 ft ladder leans against a vertical wall. The bottom slides away from the wall at 4 ft/s. How fast is the top sliding down the wall when the bottom is 7 ft from the wall?
  1. Let x be the distance from the wall to the bottom and y the height of the top. The ladder is the hypotenuse: x² + y² = 625.
    Reviewed
  2. Differentiate with respect to t: 2x dx/dt + 2y dy/dt = 0, so dy/dt = -(x/y) dx/dt.
    Reviewed
  3. \[ \left. \sqrt{625 - x^{2}} \right|_{\substack{ x=7 }} = 24 \]
    When x = 7, y = √(625 − 49).✓ Proved
  4. \[ \left. \frac{- v x}{y} \right|_{\substack{ x=7\\ y=24\\ v=4 }} = - \frac{7}{6} \]
    dy/dt = −(x/y)(dx/dt) with x = 7, y = 24, dx/dt = 4.✓ Proved
Answer \( - \frac{7}{6} \ \text{ft/s} \approx -1.167\ \text{ft/s} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.97

medium
A 5 ft tall person walks away from a 20 ft lamppost at 2 ft/s. How fast is the length of their shadow increasing?
  1. Let x be the person's distance from the post and s the shadow's length. Similar triangles: 20/(x + s) = 5/s, so s = 5x/15.
    Reviewed
  2. \[ \frac{d}{d x} \frac{x}{3} = \frac{1}{3} \]
    ds/dt = (ds/dx)(dx/dt).✓ Proved
  3. \[ \frac{2}{3} \]
    Multiply by dx/dt = 2.Reviewed
Answer \( \frac{2}{3} \ \text{ft/s} \approx 0.6667\ \text{ft/s} \)

✓ Nihil obstat Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.175

medium
Water is poured into an inverted conical tank, 9 m tall with top radius 3 m, at 6 m³/min. How fast is the water level rising when the water is 4 m deep?
  1. By similar triangles the water's radius is r = (3/9)h, so V = (1/3)π((3/9)h)²h.
    Reviewed
  2. \[ \frac{d}{d h} \frac{\pi h^{3}}{27} = \frac{\pi h^{2}}{9} \]
    dV/dt = (dV/dh)(dh/dt).✓ Proved
  3. \[ \left. \frac{54}{\pi h^{2}} \right|_{\substack{ h=4 }} = \frac{27}{8 \pi} \]
    dh/dt = (dV/dt)/(dV/dh) at h = 4.✓ Proved
Answer \( \frac{27}{8 \pi} \ \text{m/min} \approx 1.074\ \text{m/min} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 3.258

medium
Air is pumped into a spherical balloon at 16 cm³/s. How fast is the radius increasing when the radius is 4 cm?
  1. The volume of a sphere is V = (4/3)πr³.
    Reviewed
  2. \[ \frac{d}{d r} \frac{4 \pi r^{3}}{3} = 4 \pi r^{2} \]
    dV/dt = 4πr² dr/dt.✓ Proved
  3. \[ \left. \frac{4}{\pi r^{2}} \right|_{\substack{ r=4 }} = \frac{1}{4 \pi} \]
    dr/dt = (dV/dt)/(4πr²) with r = 4.✓ Proved
Answer \( \frac{1}{4 \pi} \ \text{cm/s} \approx 0.07958\ \text{cm/s} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More related rates practice

Air is pumped into a spherical balloon at 4 cm³/s. How fast is the radius increasing when the radius is 7 cm?
Problem 3.1easy✓ Every equation proved
A 10 ft ladder leans against a vertical wall. The bottom slides away from the wall at 1 ft/s. How fast is the top sliding down the wall when the bottom is 8 ft from the wall?
Problem 3.2easy✓ Nihil obstat
A stone dropped in a pond makes a circular ripple whose radius grows at 5 ft/s. How fast is the area inside the ripple growing when the radius is 8 ft?
Problem 3.3easy✓ Every equation proved
A stone dropped in a pond makes a circular ripple whose radius grows at 2 ft/s. How fast is the area inside the ripple growing when the radius is 2 ft?
Problem 3.4easy✓ Every equation proved
Car A drives east from an intersection at 23 mph and car B drives north from it at 41 mph. How fast is the distance between them changing when A is 5 mi and B is 12 mi from the intersection?
Problem 3.5easy✓ Nihil obstat
Car A drives east from an intersection at 57 mph and car B drives north from it at 40 mph. How fast is the distance between them changing when A is 3 mi and B is 4 mi from the intersection?
Problem 3.6easy✓ Nihil obstat
A 10 ft ladder leans against a vertical wall. The bottom slides away from the wall at 4 ft/s. How fast is the top sliding down the wall when the bottom is 6 ft from the wall?
Problem 3.7easy✓ Nihil obstat
Air is pumped into a spherical balloon at 16 cm³/s. How fast is the radius increasing when the radius is 10 cm?
Problem 3.8easy✓ Every equation proved
Car A drives east from an intersection at 35 mph and car B drives north from it at 26 mph. How fast is the distance between them changing when A is 6 mi and B is 8 mi from the intersection?
Problem 3.9easy✓ Nihil obstat
A 10 ft ladder leans against a vertical wall. The bottom slides away from the wall at 3 ft/s. How fast is the top sliding down the wall when the bottom is 8 ft from the wall?
Problem 3.10easy✓ Nihil obstat
A 5 ft tall person walks away from a 20 ft lamppost at 5 ft/s. How fast is the length of their shadow increasing?
Problem 3.11easy✓ Nihil obstat
A 5 ft tall person walks away from a 20 ft lamppost at 4 ft/s. How fast is the length of their shadow increasing?
Problem 3.12easy✓ Nihil obstat
Car A drives east from an intersection at 59 mph and car B drives north from it at 33 mph. How fast is the distance between them changing when A is 3 mi and B is 4 mi from the intersection?
Problem 3.13easy✓ Nihil obstat
Air is pumped into a spherical balloon at 8 cm³/s. How fast is the radius increasing when the radius is 8 cm?
Problem 3.14easy✓ Every equation proved
Water is poured into an inverted conical tank, 6 m tall with top radius 2 m, at 5 m³/min. How fast is the water level rising when the water is 4 m deep?
Problem 3.15easy✓ Nihil obstat
Air is pumped into a spherical balloon at 4 cm³/s. How fast is the radius increasing when the radius is 5 cm?
Problem 3.91medium✓ Every equation proved
A stone dropped in a pond makes a circular ripple whose radius grows at 4 ft/s. How fast is the area inside the ripple growing when the radius is 4 ft?
Problem 3.94medium✓ Every equation proved
A stone dropped in a pond makes a circular ripple whose radius grows at 2 ft/s. How fast is the area inside the ripple growing when the radius is 6 ft?
Problem 3.95medium✓ Every equation proved
A stone dropped in a pond makes a circular ripple whose radius grows at 4 ft/s. How fast is the area inside the ripple growing when the radius is 13 ft?
Problem 3.96medium✓ Every equation proved
A 13 ft ladder leans against a vertical wall. The bottom slides away from the wall at 1 ft/s. How fast is the top sliding down the wall when the bottom is 5 ft from the wall?
Problem 3.98medium✓ Nihil obstat
Car A drives east from an intersection at 52 mph and car B drives north from it at 54 mph. How fast is the distance between them changing when A is 5 mi and B is 12 mi from the intersection?
Problem 3.99medium✓ Nihil obstat
Water is poured into an inverted conical tank, 10 m tall with top radius 4 m, at 10 m³/min. How fast is the water level rising when the water is 8 m deep?
Problem 3.100medium✓ Every equation proved
Car A drives east from an intersection at 58 mph and car B drives north from it at 51 mph. How fast is the distance between them changing when A is 6 mi and B is 8 mi from the intersection?
Problem 3.101medium✓ Nihil obstat
A 10 ft ladder leans against a vertical wall. The bottom slides away from the wall at 1 ft/s. How fast is the top sliding down the wall when the bottom is 6 ft from the wall?
Problem 3.102medium✓ Nihil obstat
A stone dropped in a pond makes a circular ripple whose radius grows at 2 ft/s. How fast is the area inside the ripple growing when the radius is 14 ft?
Problem 3.103medium✓ Every equation proved
A 10 ft ladder leans against a vertical wall. The bottom slides away from the wall at 4 ft/s. How fast is the top sliding down the wall when the bottom is 8 ft from the wall?
Problem 3.104medium✓ Nihil obstat
Car A drives east from an intersection at 40 mph and car B drives north from it at 56 mph. How fast is the distance between them changing when A is 6 mi and B is 8 mi from the intersection?
Problem 3.105medium✓ Every equation proved
Air is pumped into a spherical balloon at 4 cm³/s. How fast is the radius increasing when the radius is 6 cm?
Problem 3.176medium✓ Every equation proved
Water is poured into an inverted conical tank, 9 m tall with top radius 3 m, at 10 m³/min. How fast is the water level rising when the water is 6 m deep?
Problem 3.177medium✓ Nihil obstat
A 13 ft ladder leans against a vertical wall. The bottom slides away from the wall at 3 ft/s. How fast is the top sliding down the wall when the bottom is 5 ft from the wall?
Problem 3.178medium✓ Nihil obstat
A 17 ft ladder leans against a vertical wall. The bottom slides away from the wall at 3 ft/s. How fast is the top sliding down the wall when the bottom is 8 ft from the wall?
Problem 3.179medium✓ Nihil obstat
Car A drives east from an intersection at 46 mph and car B drives north from it at 41 mph. How fast is the distance between them changing when A is 3 mi and B is 4 mi from the intersection?
Problem 3.180medium✓ Nihil obstat
Car A drives east from an intersection at 53 mph and car B drives north from it at 54 mph. How fast is the distance between them changing when A is 6 mi and B is 8 mi from the intersection?
Problem 3.181medium✓ Nihil obstat
A 5 ft ladder leans against a vertical wall. The bottom slides away from the wall at 3 ft/s. How fast is the top sliding down the wall when the bottom is 3 ft from the wall?
Problem 3.182medium✓ Nihil obstat
Car A drives east from an intersection at 47 mph and car B drives north from it at 27 mph. How fast is the distance between them changing when A is 6 mi and B is 8 mi from the intersection?
Problem 3.215medium✓ Nihil obstat
Car A drives east from an intersection at 38 mph and car B drives north from it at 54 mph. How fast is the distance between them changing when A is 5 mi and B is 12 mi from the intersection?
Problem 3.216medium✓ Nihil obstat
A 10 ft ladder leans against a vertical wall. The bottom slides away from the wall at 2 ft/s. How fast is the top sliding down the wall when the bottom is 6 ft from the wall?
Problem 3.217medium✓ Nihil obstat
Water is poured into an inverted conical tank, 6 m tall with top radius 2 m, at 9 m³/min. How fast is the water level rising when the water is 2 m deep?
Problem 3.218medium✓ Nihil obstat
Air is pumped into a spherical balloon at 16 cm³/s. How fast is the radius increasing when the radius is 7 cm?
Problem 3.219medium✓ Every equation proved
A stone dropped in a pond makes a circular ripple whose radius grows at 3 ft/s. How fast is the area inside the ripple growing when the radius is 2 ft?
Problem 3.220medium✓ Every equation proved
Water is poured into an inverted conical tank, 9 m tall with top radius 3 m, at 5 m³/min. How fast is the water level rising when the water is 3 m deep?
Problem 3.221medium✓ Nihil obstat
Car A drives east from an intersection at 59 mph and car B drives north from it at 60 mph. How fast is the distance between them changing when A is 3 mi and B is 4 mi from the intersection?
Problem 3.222medium✓ Nihil obstat
Car A drives east from an intersection at 22 mph and car B drives north from it at 41 mph. How fast is the distance between them changing when A is 9 mi and B is 12 mi from the intersection?
Problem 3.255medium✓ Nihil obstat
Air is pumped into a spherical balloon at 8 cm³/s. How fast is the radius increasing when the radius is 6 cm?
Problem 3.256medium✓ Every equation proved
A stone dropped in a pond makes a circular ripple whose radius grows at 1 ft/s. How fast is the area inside the ripple growing when the radius is 9 ft?
Problem 3.257medium✓ Every equation proved
A stone dropped in a pond makes a circular ripple whose radius grows at 4 ft/s. How fast is the area inside the ripple growing when the radius is 18 ft?
Problem 3.259medium✓ Every equation proved
Air is pumped into a spherical balloon at 16 cm³/s. How fast is the radius increasing when the radius is 8 cm?
Problem 3.260medium✓ Every equation proved
Car A drives east from an intersection at 21 mph and car B drives north from it at 25 mph. How fast is the distance between them changing when A is 9 mi and B is 12 mi from the intersection?
Problem 3.261medium✓ Nihil obstat
Air is pumped into a spherical balloon at 12 cm³/s. How fast is the radius increasing when the radius is 8 cm?
Problem 3.262medium✓ Every equation proved
Car A drives east from an intersection at 54 mph and car B drives north from it at 22 mph. How fast is the distance between them changing when A is 6 mi and B is 8 mi from the intersection?
Problem 3.295medium✓ Nihil obstat
Air is pumped into a spherical balloon at 100 cm³/s. How fast is the radius increasing when the radius is 2 cm?
Problem 3.296medium✓ Every equation proved
Car A drives east from an intersection at 20 mph and car B drives north from it at 38 mph. How fast is the distance between them changing when A is 6 mi and B is 8 mi from the intersection?
Problem 3.297medium✓ Nihil obstat
A 25 ft ladder leans against a vertical wall. The bottom slides away from the wall at 3 ft/s. How fast is the top sliding down the wall when the bottom is 7 ft from the wall?
Problem 3.298medium✓ Nihil obstat
Air is pumped into a spherical balloon at 4 cm³/s. How fast is the radius increasing when the radius is 10 cm?
Problem 3.299medium✓ Every equation proved
A 6 ft tall person walks away from a 12 ft lamppost at 4 ft/s. How fast is the length of their shadow increasing?
Problem 3.300medium✓ Every equation proved
Air is pumped into a spherical balloon at 8 cm³/s. How fast is the radius increasing when the radius is 4 cm?
Problem 3.301medium✓ Every equation proved
Car A drives east from an intersection at 22 mph and car B drives north from it at 57 mph. How fast is the distance between them changing when A is 5 mi and B is 12 mi from the intersection?
Problem 3.302medium✓ Nihil obstat
Car A drives east from an intersection at 54 mph and car B drives north from it at 49 mph. How fast is the distance between them changing when A is 9 mi and B is 12 mi from the intersection?
Problem 3.335medium✓ Nihil obstat
A 6 ft tall person walks away from a 18 ft lamppost at 3 ft/s. How fast is the length of their shadow increasing?
Problem 3.336medium✓ Nihil obstat
A stone dropped in a pond makes a circular ripple whose radius grows at 2 ft/s. How fast is the area inside the ripple growing when the radius is 10 ft?
Problem 3.337medium✓ Every equation proved
A 25 ft ladder leans against a vertical wall. The bottom slides away from the wall at 1 ft/s. How fast is the top sliding down the wall when the bottom is 7 ft from the wall?
Problem 3.338medium✓ Nihil obstat
A 6 ft tall person walks away from a 12 ft lamppost at 2 ft/s. How fast is the length of their shadow increasing?
Problem 3.339medium✓ Every equation proved
A stone dropped in a pond makes a circular ripple whose radius grows at 3 ft/s. How fast is the area inside the ripple growing when the radius is 9 ft?
Problem 3.340medium✓ Every equation proved
Water is poured into an inverted conical tank, 9 m tall with top radius 3 m, at 3 m³/min. How fast is the water level rising when the water is 7 m deep?
Problem 3.341medium✓ Every equation proved
Car A drives east from an intersection at 41 mph and car B drives north from it at 25 mph. How fast is the distance between them changing when A is 3 mi and B is 4 mi from the intersection?
Problem 3.342medium✓ Nihil obstat