Related rates practice problems
Ladders, balloons, tanks and shadows: how one rate of change drives another. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.
Air is pumped into a spherical balloon at 4 cm³/s. How fast is the radius increasing when the radius is 7 cm?
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?
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?
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?
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?
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?
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?
Air is pumped into a spherical balloon at 16 cm³/s. How fast is the radius increasing when the radius is 10 cm?
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?
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?
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?
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?
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?
Air is pumped into a spherical balloon at 8 cm³/s. How fast is the radius increasing when the radius is 8 cm?
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?
Air is pumped into a spherical balloon at 4 cm³/s. How fast is the radius increasing when the radius is 5 cm?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?
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?