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In order to solve problems involving distance, it is necessary to

A typical problem involving distance and the formula d = rt is usually entitled a uniform motion problem. The problem will have something to do with objects moving at a constant rate of speed or an average rate of speed.

Suppose two sisters live 240 miles apart. One sister has three young children who are planning to visit their aunt for a week. To prevent driving so far, the sisters agree to leave at the same time, drive toward each other, and meet somewhere along the route. The sister with the three children tends to drive carefully and obey the speed limit. Her average rate of speed is 70 mph. The other sister drives too fast, and her average rate of speed is 80 mph. How long will it take the two sisters to meet each other to transfer the children?

First, notice that two cars are traveling toward each other. Their average rates of speed are given. The total distance is given, and, even though we do not know the time the cars began to travel, we are told that the two cars did leave home at the same time.

Two tools are very helpful in writing an equation for a uniform motion problem: diagrams and charts.

A typical diagram to represent this data would look like the following:

 One Sister’s Home Second Sister’s Home 240 miles

This indicates that sister one is traveling from her home toward sister two. Sister two, likewise, is traveling from her home toward sister one. Together, they will travel a total distance of 240 miles - the distance from one home to the other.

A chart would be used to include all the data necessary for writing an equation. The chart (shown below) uses the formula distance = rate times time (d = rt). In the chart below, sister one is the sister with the children who drives at 70 miles per hour, and sister two is the sister driving to pick up the children. She drives 80 miles per hour. The speeds are indicated in the rate column. The amount of time they travel is unknown and is designated by the letter t in the time column. The distance column contains the product of the rate and the time.

 time distance Sister 1 70 mph t 70t Sister 2 80 mph t 80t total distance = 240 miles

Using the chart and the diagram, we now write the equation. The last column of the chart tells us the total distance for each sister, and the diagram tells us the total distance driven by the two sisters together. Combining these two ideas we have:

 (distance of Sister 1) plus (distance of Sister 2) equals (total distance) 70t + 80t = 240

We are now ready to solve the equation:   The result indicates that it will take 1 and hour for the two sisters to meet. Therefore, it will take 1 hour and 36 minutes for them to meet. Let's Practice

 Question #1 Two friends leave a hotel at the same time traveling in opposite directions. They travel for four hours and are then 480 miles apart. If Susan travels 10 miles per hour faster than Joan, find the average rate of speed for each person.

 Question #2 A runner decides to run out in the country. He begins to run at an average rate of 9 miles per hour. He runs a certain distance and then turns around and returns along the same route at an average rate of 6 miles per hour. If the round trip took 2 and a half hours, how far did the runner travel before turning around? Try These
 Question #1  Two students on bicycles leave their classroom building at 10:00 AM and travel in opposite directions. If the average speed of one of the students is 12 kilometers per hour and the average speed of the other student is 14 kilometers per hour, at what time will they be 65 kilometers apart? A. 2.5 hours B. 12:30 PM C. 7:30 AM D. 32.5 hours

 Question #2  James leaves his home town traveling 70 miles per hour. At the same time Paul leaves home traveling 75 miles per hour. The two live 580 miles apart and are traveling to meet each other for a lunch meeting at noon. What time did they leave their homes? A. 4:00 PM B. 4 hrs C. 8:00 AM D. 116 hours

As you can see, this type of problem requires carefully constructing a diagram, a chart, and an equation. Following the writing of the equation, you must carefully solve for the assigned variable. After solving the equation, you must be careful to answer the question that was asked in the problem. You should always take time also to check your solution in the chart and in the equation that you wrote.

D Saye

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