## Related Rates

### Related Rates: Problems and Solutions

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:

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?

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

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?

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?

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

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

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 $x=5.$

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

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