Section 2.8

Related rates

Often, when an object moves, more than one dimension changes. For example, if a ladder is sliding down a wall, the top goes down and the base goes to the side. Here’s an animation on Geogebra:

https://www.geogebra.org/m/Yfudjzga

Or when a balloon is inflating, for example, the volume changes at one rate while the radius changes as well!

Example question #

Air is blown in to a spherical balloon at a constant rate of $15 \text{in}^3/\text{s}$. How fast is the radius expanding when the radius is 1 in? 6in? 12in?

… we’ll do the solution shortly.

Solving These Problems - the process #

1. Read carefully. Ask “what is the problem looking for?”
2. Draw a diagram if possible.
3. Give symbols to known and unknown quantities.
4. Write an equation to relate our quantities. You may need to use geometry.
5. Use the chain rule / implicit differentiation with respect to time.
6. Substitute information into the derivative equation and solve for the unknown.

Example #

Air is blown in to a spherical balloon at a constant rate of $15 \text{in}^3/\text{s}$. How fast is the radius expanding when the radius is 1 in? 6in? 12in?

The fact that as the radius gets bigger the balloon is expanding less ($dr/dt$ is getting smaller) makes sense from experience blowing balloons. If you haven’t seen a balloon inflated in a long time here’s an entire youtube channel of some guy blowing up thousands of balloons, because Internet.

Example #

A 10ft ladder is leaning against a wall. Its base is sliding away from the wall at 1ft/s. How fast is the top of the ladder sliding down when the base is 6ft from the wall?

Example #

A (circular cylindrical) cone with a base radius of 2.5cm and a height of 10cm is being filled, slowly, with delicious ice cream at a rate of $2\text{cm}^3/\text{min}$. Find the rate at which the height of ice cream is rising when the height is 8 cm.

Example #

Mario and Luigi start in the same location in their Karts. Mario is going east (towards the finish line) at a rate of 95km/h. Luigi, who isn’t good at racing, goes south (not towards the finish line) at 105km/h.

At what rate are Mario and Luigi driving away from each other after 2 hours?

Example #

An actor walks in a straight path at a speed of 2m/s. A spotlight is mounted 20m from the straight path the actor is walking on the stage and is concentrated on the actor. At what rate is the spotlight moving when the man is 4m from the point on the straightway closest to the search light?

Google didn’t like my input, but WolframAlpha comes to save the day. Here’s the input so you can see the calculation.

By the way, that works to 5.5 degrees per second.