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**Word Problems - Motion By Joe Joyner Math 04 Intermediate Algebra**

Link to Practice Problems

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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.

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**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”

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**Rate, Time, and Distance Problems**

is given by the “distance, rate, time” formula: d = rt.

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**Rate, Time, and Distance Problems**

Example 1 You ride your bike for 7 hours. If you travel miles, what is your average speed?

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**Rate, Time, and Distance Problems**

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

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**Rate, Time, and Distance Problems**

Example 1 You can use a spreadsheet (Excel, for example) to build a model for this problem.

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**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.

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**Rate, Time, and Distance Problems**

Example 1 Represent the variable rate with r . You can use the distance, rate, time formula. d = rt

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**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.

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**Rate, Time, and Distance Problems**

Example 1 is our mathematical model. Some mathematical models can be easy!

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**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

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**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.

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**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 AM on the same day as Dan and travels at mph in the opposite direction as the route taken by Dan.

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**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?

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**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.

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**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.

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**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

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**Rate, Time, and Distance Problems**

Example 2 You can use a spreadsheet to build a model for this problem too.

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**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.

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**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

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**Rate, Time, and Distance Problems**

Example 2 55(t+1) + 45t = 280 55t t = 280 100t + 55 = 280 100t = 225 t = 2.25 hours

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**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?

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**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

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**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

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**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

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**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 = (3.25) d = 6.5 miles

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**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.

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Explore

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**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

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DISTANCE = RATE*TIME D = rt D = r(t) D = r x t.

DISTANCE = RATE*TIME D = rt D = r(t) D = r x t.

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