Two quantities tied together by a formula have their rates tied together too. Differentiate the formula with respect to time, put the numbers in last, and the unknown rate falls out.
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A ship sails east from a harbor at 6 km/h. Another sails north from the same harbor at 9 km/h. At the moment when they are 3 km and 4 km from the harbor, how fast is the distance between them growing?
A balloon rises straight up at 11 ft/s from a point 1500 ft from an observer. How fast is its distance from the observer growing when it is 800 ft up?
A 13 m ladder leans against a wall. The foot is pulled away at 1 m/s. How fast is the top sliding down when the foot is 5 m from the wall?
The area of a circle grows at 14 cm2/s. How fast is the radius growing when r=8 cm?
A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 4 cm?
A person 5 ft tall walks away from a 15 ftlamppost at 2 ft/s. How fast is the length of the shadow changing?
A balloon's radius grows at 1 cm/s. How fast is its volume growing when the radius is 6 cm?
A 10 ft ladder leans against a wall. Its foot slides away at 1 ft/s. How fast is the top sliding down when the foot is 6 ft from the wall?
A 15 m ladder leans against a wall. The foot is pulled away at 1 m/s. How fast is the top sliding down when the foot is 9 m from the wall?
A person 6 ft tall walks away from an 18 ftlamppost at 3 ft/s. How fast is the tip of the shadow moving?
A ship sails east from a harbor at 9 km/h. Another sails north from the same harbor at 9 km/h. At the moment when they are 8 km and 15 km from the harbor, how fast is the distance between them growing?
A person 5 ft tall walks away from a 12 ftlamppost at 4 ft/s. How fast is the tip of the shadow moving?
A ship sails east from a harbor at 4 km/h. Another sails north from the same harbor at 2 km/h. At the moment when they are 9 km and 12 km from the harbor, how fast is the distance between them growing?
Solve harder related rates problems
Car A is 6 mi east of an intersection driving toward it at 45 mph; car B is 8 mi north driving toward it at 49 mph. How fast is the distance between them changing?
A plane flies at an altitude of 7 km at 243 km/h, directly away from an observer. How fast is the angle of elevation changing when the horizontal distance is 7 km?
Water drains from a cylindrical tank of radius 5 m at 2 m3/min. How fast is the water level dropping?
A plane flies at an altitude of 6 km at 106 km/h, directly away from an observer. How fast is the angle of elevation changing when the horizontal distance is 12 km?
Water drains from a cylindrical tank of radius 5 m at 4 m3/min. How fast is the water level dropping?
Car A is 8 mi east of an intersection driving toward it at 47 mph; car B is 6 mi north driving toward it at 52 mph. How fast is the distance between them changing?
Car A is 8 mi east of an intersection driving toward it at 37 mph; car B is 6 mi north driving toward it at 32 mph. How fast is the distance between them changing?
A ship sails east from a harbor at 6 km/h. Another sails north from the same harbor at 9 km/h. At the moment when they are 3 km and 4 km from the harbor, how fast is the distance between them growing?554 km/h
A balloon rises straight up at 11 ft/s from a point 1500 ft from an observer. How fast is its distance from the observer growing when it is 800 ft up?1788 ft/s
A 13 m ladder leans against a wall. The foot is pulled away at 1 m/s. How fast is the top sliding down when the foot is 5 m from the wall?125 m/s
The area of a circle grows at 14 cm2/s. How fast is the radius growing when r=8 cm?8π7 cm/s
A balloon's radius grows at 2 cm/s. How fast is its volume growing when the radius is 4 cm?128π cm3/s
A person 5 ft tall walks away from a 15 ftlamppost at 2 ft/s. How fast is the length of the shadow changing?1 ft/s
A balloon's radius grows at 1 cm/s. How fast is its volume growing when the radius is 6 cm?144π cm3/s
A 10 ft ladder leans against a wall. Its foot slides away at 1 ft/s. How fast is the top sliding down when the foot is 6 ft from the wall?43 ft/s
A 15 m ladder leans against a wall. The foot is pulled away at 1 m/s. How fast is the top sliding down when the foot is 9 m from the wall?43 m/s
A person 6 ft tall walks away from an 18 ftlamppost at 3 ft/s. How fast is the tip of the shadow moving?29 ft/s
A ship sails east from a harbor at 9 km/h. Another sails north from the same harbor at 9 km/h. At the moment when they are 8 km and 15 km from the harbor, how fast is the distance between them growing?17207 km/h
A person 5 ft tall walks away from a 12 ftlamppost at 4 ft/s. How fast is the tip of the shadow moving?748 ft/s
A ship sails east from a harbor at 4 km/h. Another sails north from the same harbor at 2 km/h. At the moment when they are 9 km and 12 km from the harbor, how fast is the distance between them growing?4 km/h
Solve harder related rates problems
Car A is 6 mi east of an intersection driving toward it at 45 mph; car B is 8 mi north driving toward it at 49 mph. How fast is the distance between them changing?−5331 mph
A plane flies at an altitude of 7 km at 243 km/h, directly away from an observer. How fast is the angle of elevation changing when the horizontal distance is 7 km?−14243 rad/h
Water drains from a cylindrical tank of radius 5 m at 2 m3/min. How fast is the water level dropping?25π2 m/min
A plane flies at an altitude of 6 km at 106 km/h, directly away from an observer. How fast is the angle of elevation changing when the horizontal distance is 12 km?−1553 rad/h
Water drains from a cylindrical tank of radius 5 m at 4 m3/min. How fast is the water level dropping?25π4 m/min
Car A is 8 mi east of an intersection driving toward it at 47 mph; car B is 6 mi north driving toward it at 52 mph. How fast is the distance between them changing?−5344 mph
Car A is 8 mi east of an intersection driving toward it at 37 mph; car B is 6 mi north driving toward it at 32 mph. How fast is the distance between them changing?−5244 mph
A circle's radius grows at 3 cm/s. How fast is its area growing when the radius is 5 cm?
The area is pi r squared.
Differentiating with respect to time gives 2 pi r times dr/dt.
Putting in r equals 5 and dr/dt equals 3 gives 30 pi square centimeters per second.
The answer is 30π cm2/s.
Where students go wrong
Substituting the numbers before differentiating. Put the 5 in first and the radius becomes a constant, its derivative is zero, and the answer comes out zero as well. The numbers go in after the differentiating, never before.
Also called rates of change word problems, ladder problem, expanding balloon, cone filling with water or shadow problem.
What you can put on this worksheet
Related rates
A square's side grows at 5 cm/s. How fast is its area growing when the side is 10 cm?→ 100 cm2/s
Angles, draining tanks, cars approaching
Car A is 5 mi east of an intersection driving toward it at 50 mph; car B is 12 mi north driving toward it at 23 mph. How fast is the distance between them changing?→ −13526 mph
Run the other way round, and shadows
A person 6 ft tall walks away from an 18 ftlamppost at 2 ft/s. How fast is the length of the shadow changing?→ 1 ft/s
Questions about these worksheets
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Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
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That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the related rates questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 3: related rates, angles, draining tanks, cars approaching and run the other way round, and shadows. Tick as many as you want and set how many of each, or let it spread them evenly.
Substituting the numbers before differentiating. Put the 5 in first and the radius becomes a constant, its derivative is zero, and the answer comes out zero as well. The numbers go in after the differentiating, never before.