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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:

Click to view to see the four subcategories.
1. Geometric fact. Typical problems: A circle’s radius changes, a snowball melts, a rectangle’s height and/or width changes.
2. Trig function. Typical problems: A searchlight rotates, a rocket takes off, a kite travels horizontally.
3. Similar triangles. Water fills a cone or trough, sand falls onto a conical pile, person walks away from a light pole that casts a shadow.
4. Pythagorean theorem. Typical problems: Cars/ships/joggers move along 90 degree paths, baseball players run along the diamond, boat is pulled toward a dock.

We have an example of each type below.

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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?

Click to view full Calculus solution to “Snowball melts.”

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?

Click to view full Calculus solution to “At what rate does the angle change as a ladder slides away from a house?”


4. How fast is the water level falling as water drains from the cone? (uses Similar Triangles)
water draining from a cone is a standard related rates problem.
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?

Click to view full Calculus solution to “How fast is the water level falling as water drains from the cone?”

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 $x=5.$

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

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