Related Rates!?

Question

I have to write a story about 2 kids trick-or-treating. How would i make a calculus stopry out of it?

I thought i could do something like .
There were 54 house in the neighborhood.
Mike traveled 4 house / 5 minutes
Jake traveled 2 house / 5 minutes.

What other information do i need to add to the story?
What should the question be?
How would u slove this?

Thanks!

Answer 1

I think it's going to be difficult for you to come up with a related rate if you deal with how many houses they've covered. Usually you want some sort of geometric relationship in order to setup your related rates problems.

Trick-or-treat is at night. There are street lights at night. What if you did the classic problem about someone walking away from a streetlight with a growing shadow. You could say that Mike is 4 feet tall and Jake is 4.5 feet tall. Assuming that they walk away from the streetlight at the same speed, compare the rate that Mike's shadow grows with the rate that Jake's shadow grows.

The tip of each shadow is the endpoint of a line that goes through the top of the head of the boy and the top of the streetlight. This is the hypotenuse of the triangle with the streetlight on one leg and the distance from the streetlight to the boy as well as the length of the shadow on the other leg. There is a second triangle with the boy as one leg and the length of the shadow on the other leg. The two triangles are similar. That is, if you know the ratio of one leg to the other on the big triangle, you'll get the same ratio for the legs of the small triangle.

So basically, you'll have an equation which relates x (where the boy is) with l (the length of the triangle). If you take the derivative of this relationship with respect to time, you'll get a relationship between dx/dt and dl/dt. So if you know dx/dt (the boy's speed) you'll be able to find dl/dt.

So all you need is:

*) The height of the two boys
*) The height of the stoplight
*) The rate that they're running away from the streetlight

And you can calculate the rate of change of the length of the streetlight for each boy. I think you'll find that the taller boy's shadow grows faster than the shorter boy's.

You can setup lots of other similar problems dealing with shadows and light. Use your imagination. Setup the geometric relationships. Take the derivatives to get your related rates.