Related Rates (Calculus I)
Free printable Calculus I worksheet on related rates: expanding circles and spheres, sliding ladders, melting cubes and filling conical tanks. Set up the relation, differentiate with the chain rule, solve for the unknown rate. Every answer computed and exact.
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Calculus I · Math worksheet
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Math
Related Rates
For each situation, write the relation between the quantities, differentiate both sides with respect to time t (chain rule), substitute the known values, and solve for the unknown rate. Include units.
- 1.The radius of a circular oil slick grows at a constant 3 cm/min. How fast is the area increasing at the instant the radius is 4 cm? Give an exact answer in terms of π.
- 2.A spherical balloon is inflated so its radius increases at 3 cm/s. How fast is the volume increasing when the radius is 6 cm? Give an exact answer in terms of π.
- 3.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 of the ladder sliding down the wall when the bottom is 6 ft from the wall?
- 4.Each edge of a cube of ice melts (shrinks) at 3 mm/min. How fast is the volume decreasing when each edge is 2 mm?
- 5.Water is poured into a conical tank so the water level rises at 2 cm/min. At every depth the water surface's radius is half the water's depth. How fast is the volume of water increasing when the depth is 10 cm? Give an exact answer in terms of π.
- 6.The radius of a circular oil slick grows at a constant 1 cm/min. How fast is the area increasing at the instant the radius is 10 cm? Give an exact answer in terms of π.
- 7.A 25 ft ladder leans against a vertical wall. The bottom slides away from the wall at 2 ft/s. How fast is the top of the ladder sliding down the wall when the bottom is 7 ft from the wall?
- 8.A spherical balloon is inflated so its radius increases at 3 cm/s. How fast is the volume increasing when the radius is 4 cm? Give an exact answer in terms of π.
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