Application Problem Lecture Notes

Just a few quick notes for application problems:

When most students think of motion problems, they think of rate ´ time = distance. This is true when rate is measured as a unit of distance divided by a unit of time such as miles per hour. Sometimes rate is not measured that way. Look at the department’s Motion Problems handout, number 8. You’ll notice they talk about a cross-country runner averaging 10 miles per hour on level ground and 6 miles per hour on hilly ground. If you’ve ever been a runner or been around runners, you’ll know that this is not really the way they talk. A runner would say that the cross-country runner in this problem averages 6 minutes per mile on level ground and 10 minutes per mile on hilly ground. Let’s work this problem with the minutes per mile units.

First, I’ll set up a chart: (These kind of charts come in handy for motion, mixture and interest rate problems.) Note that with rate measured as minutes per mile that translates to r = t/d or rate ´ distance = time.

Given info is in black; other info is in red.

	rate	´ distance	= time
level	6 min/mile	12 -x	6(12 –x)
hilly	10 min/mile	x	10x
		12 miles	100 minutes

I let x stand for the number of hilly miles since that is what the problem is asking us to find. You’ll notice that the last column filled in itself by using the formula. Whenever I use these charts to solve problems, I fill in two of the three columns and let the formula fill in the last column. Now I should have one left over piece of information that I can use to set up my equation. In this case that piece of information is evident from the chart. The time on level ground and hilly ground should add up to the 100 minutes: 6(12 –x) + 10x = 100. Solve for x and we can answer the question. x = 7 miles.

You can practice doing this problem again as it is originally written. You’ll notice that since the rate is in miles per hour you get r = d/t or r ´ t = d.

The point of this problem is don’t just blindly follow formulas that you may have memorized, but use your head and look at the units to figure out what is going on. Use the skills that you practiced when we did unit anaylsis.