We will solve every Related Rates problem using the same Problem Solving Strategy time and again. You can see an overview of that strategy here (link will open in a new tab).

As stated in the Problem Solving Strategy, nearly every Related Rates problem will fall into one of four subcategories. With practice, you’ll learn to recognize each:

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1. Snowball melts at given rate (uses Geometric Fact)

A spherical snowball melts at the rate of $2 \pi$ cm$^3$/hr. It melts symmetrically such that it is always a sphere. How fast is its radius changing at the instant $r = 10$ cm?

2. Snowball melts, area decreases at given rate (uses Geometric Fact)

A spherical snowball melts symmetrically such that it is always a sphere. Its surface area decreases at the rate of $\pi$ in$^2$/min. How fast is its radius changing at the instant when $r = 2$ inches?

Click to view Calculus solution to “Snowball melts, area decreases at given rate,” with video

3. At what rate does the angle change as a ladder slides away from a house? (uses Trig function)

A 10-ft ladder leans against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house at 2 ft/s. At what rate is the angle between the ladder and the ground changing when the base is 8 ft from the house?

4. How fast is the water level falling as water drains from the cone? (uses Similar Triangles)

An inverted cone is 20 cm tall, has an opening radius of 8 cm, and was initially full of water. The water now drains from the cone at the constant rate of 15 cm$^3$ each second. The water’s surface level falls as a result. At what rate is the water level falling when the water is halfway down the cone?

To access more problems and solutions, including AP-style multiple choice questions, log in for free with your Google, Apple or Facebook account, or create a dedicated Matheno in 60 seconds. You’ll also then be able to mark the problems *you* want to be sure to review before your exams. Simply use the log-in area at the bottom of this screen.

5. Lamp post casts a shadow of a man walking. (uses Similar Triangles)

A 1.8-meter tall man walks toward a 6.0-meter lamp post at the rate of 1.5 m/s. The light at the top of the post casts a shadow behind the man. How fast is the “head” of his shadow moving along the ground?

Click to view Calculus solution to “Lamp post casts a shadow of a man walking,” with video

6. How fast is the ladder's top sliding? (uses Pythagorean Theorem)

A 10-ft ladder is leaning against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house. When the base has slid to 8 ft from the house, it is moving horizontally at the rate of 2 ft/sec. How fast is the ladder’s top sliding down the wall when the base is 8 ft from the house?

Click to view full Calculus solution to “How fast is the ladder’s top sliding?” with video

7. Given an equation, find a rate.

Not all Related Rates problems are word problems. Here’s a different type that came from a student. The problem gives you an equation, and then asks you to find a rate:

If $y = x^3 + 2x$ and $\dfrac{dx}{dt} = 6$, find $\dfrac{dy}{dt}$ when $x=5.$

If $y = x^3 + 2x$ and $\dfrac{dx}{dt} = 6$, find $\dfrac{dy}{dt}$ when $x=5.$

Click to view full Calculus solution to “Given an equation, find a rate.”

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