Struggling through a related rates problem

We’ve been talking about ideas in the related rates chapter in our calculus book for the last few days. My son has struggled with the ideas in this chapter much more than I expected he would. Fortunately many folks on twitter who have much more experience teaching calculus than I do have told me that this section often gives students a lot of trouble.

Over the last few days my son had worked through maybe 15 of the problems in our book – so I just picked one that he hadn’t done yet. Here’s the problem:

Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2^o per minute. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60^o

Here’s the introduction to the problem and his initial thoughts:

(oh, and I should say at the outset, we’d not looked at this problem before and I didn’t realize that we needed a calculator, so I just banged out a few numbers on my computer just to speed things along. Most of my calculations were right, but one at the end of the 2nd video was wrong. Whatever I typed in produced the answer of 0.2 m / min, but if I’d actually typed in what we had on the board I would have found 0.4 m / min. That typo by me led to a little confusion in the 3rd video when we went to check our answer. Sorry about that error.)

Here’s the next part of his work.

There’s one part of this problem that we’d not really talked about carefully yet -> the angle numbers in the problem are given in degrees, but in calculus you need to be using radians. Since the main focus of this problem / session is related rates, I just explained that fact quickly in the video.

Again, sorry for the typo by me in evaluating the value of the answer.  Fortunately we decided to check the answer – which was also an important conceptual calculus exercise.

Here we checked the answer from the last video and found the change was double what we were expecting. This was unexpected!

We checked the prior math off camera and found the typo. Luckily it was easy to find since we’d filmed the project. Hopefully this was a nice (and accidental) way to show that checking your answers is important 🙂