Word Problems - Motion By Joe Joyner Math 04 Intermediate Algebra Link to Practice Problems

Introduction In this module, you’ll continue to develop and work with mathematical models. When solving practical application problems, you try to find a mathematical model for the problem. A mathematical model does not necessarily have to be complicated. It can be relatively simple. This is usually the case when only one or two variables are required to build a linear model. Let’s begin.

Rate, Time, and Distance Problems If an object such as an automobile or an airplane travels at a constant, or uniform, rate of speed, “r” , then the distance traveled by the object, “d”, during a period of time, “t”

Rate, Time, and Distance Problems is given by the “distance, rate, time” formula: d = rt.

Rate, Time, and Distance Problems Example 1 You ride your bike for 7 hours. If you travel 36.75 miles, what is your average speed?

Rate, Time, and Distance Problems Example 1 The quantities in this problem are: distance (constant at 36.75 miles), time (constant at 7 hours), and rate, or speed (unknown variable).

Rate, Time, and Distance Problems Example 1 You can use a spreadsheet (Excel, for example) to build a model for this problem.

Rate, Time, and Distance Problems Example 1 Explore To access the spreadsheet, click the word Explore. Then explore with the rate to see if you can solve the problem.

Rate, Time, and Distance Problems Example 1 Represent the variable rate with r . You can use the distance, rate, time formula. d = rt

Rate, Time, and Distance Problems Example 1 But since you know the distance and time, and wish to solve for rate, it would be helpful to solve the equation for r first.

Rate, Time, and Distance Problems Example 1 is our mathematical model. Some mathematical models can be easy!

Rate, Time, and Distance Problems Example 1 Now we can solve for the rate, r , by dividing the distance by the time. 5.25 miles per hour

Rate, Time, and Distance Problems When you read a word problem that involves rate, time, and distance, note whether the problem situation involves motion in the same direction; motion in opposite directions; a round trip.

Rate, Time, and Distance Problems Example 2 Dan and Emily are truck drivers. Dan, averaging 55 miles per hour (mph), begins a 280-mile trip from their company’s Norfolk warehouse to Charlotte, NC at 7 AM. Emily sets out from the Charlotte warehouse at 8 AM on the same day as Dan and travels at 45 mph in the opposite direction as the route taken by Dan.

Rate, Time, and Distance Problems Example 2 How many hours will Emily have been driving when she and Dan pass each other? How will you start to set up a model for solving this problem?

Rate, Time, and Distance Problems Example 2 What is the variable that you must solve for? time Is the length of time traveled the same for Dan and Emily when they pass each other? No.

Rate, Time, and Distance Problems Example 2 Why is the time different for the two drivers? Dan started at 7 AM and Emily started at 8 AM. Dan averaged 55 mph and Emily averaged 45 mph.

Rate, Time, and Distance Problems Example 2 Let t represent the amount of time that Emily travels until the trucks pass each other. In terms of t , how long will Dan have been on the road when the trucks pass each other? One hour longer or ... t + 1

Rate, Time, and Distance Problems Example 2 You can use a spreadsheet to build a model for this problem too.

Rate, Time, and Distance Problems Explore Example 2 To access the spreadsheet, click the word Explore. Then explore with Emily’s time to see if you can solve the problem.

Rate, Time, and Distance Problems Example 2 The mathematical model for this problem is: Dan’s Distance + Emily’s Distance = 280 miles Dan’s rate*Dan’s time + Emily’s rate*Emily’s time = 280 55(t+1) + 45t = 280

Rate, Time, and Distance Problems Example 2 55(t+1) + 45t = 280 55t+55 + 45t = 280 100t + 55 = 280 100t = 225 t = 2.25 hours

Rate, Time, and Distance Problems Example 3 Jason and LeRoy are entered in a 26-mile marathon race. Jason’s average pace is 6 miles per hour (mph) and LeRoy’s average pace is 8 mph. Both runners start at the same time. How far from the finish line will Jason be when LeRoy crosses the finish line?

Rate, Time, and Distance Problems Example 3 What are the known constants? Jason’s rate of 6 mph LeRoy’s rate of 8 mph Race distance of 26 miles

Rate, Time, and Distance Problems Example 3 What are the unknowns? The amount of time it takes LeRoy to finish the race The distance Jason has to run when LeRoy finishes

Rate, Time, and Distance Problems Example 3 Let LeRoy’s time be t . What is the distance, rate, time, model for Leroy in this problem? 8t = 26 What is the solution for t ? t = 3.25 hours

Rate, Time, and Distance Problems Example 3 At the time that LeRoy crosses the finish line, Jason has run for the same amount of time, t . What is the model for how far Jason is from the finish line at that time? d = 26 - 6(3.25) d = 6.5 miles

Rate, Time, and Distance Problems Do you think you’ve got the concept of solving motion (rate, time distance) problems? Look at the next slide. If you want to try the interactive web site that the slide came from, click on the word Explore to go there.

Explore

Rate, Time, and Distance Problems Hopefully, you are now ready to practice motion problems for yourself. When you click the Go To Practice Problems link below, your web browser will open the practice problem set. Go To Practice Problems