ElementaryAlgebra2e-WEB_3zxfu3Z-75

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• Create a table to organize the information.
• Label the columns “Rate,” “Time,” and “Distance.”
• List the two scenarios.
• Write in the information you know.
Step 2. Identify what we are looking for.
• We are asked to find the speed of both trains.
• Notice that the distance formula uses the word “rate,” but it is more common to use “speed” when we talk about
vehicles in everyday English.
Step 3. Name what we are looking for. Choose a variable to represent that quantity.
• Complete the chart
• Use variable expressions to represent that quantity in each row.
• We are looking for the speed of the trains. Let’s let r represent the speed of the local train. Since the speed of the
express train is 12 mph faster, we represent that as
Fill in the speeds into the chart.
Multiply the rate times the time to get the distance.
Step 4. Translate into an equation.
• Restate the problem in one sentence with all the important information.
• Then, translate the sentence into an equation.
• The equation to model this situation will come from the relation between the distances. Look at the diagram we
drew above. How is the distance travelled by the express train related to the distance travelled by the local train?
• Since both trains leave from Pittsburgh and travel to Washington, D.C. they travel the same distance. So we write:
3.5 • Solve Uniform Motion Applications 361
Step 5. Solve the equation using good algebra techniques.
Now solve this equation.
So the speed of the local train is 48 mph.
Find the speed of the express train.
The speed of the express train is 60 mph.
Step 6. Check the answer in the problem and make sure it makes sense.
express train
local train
Table 3.11
Step 7. Answer the question with a complete sentence.
• The speed of the local train is 48 mph and the speed of the express train is 60 mph.
TRY IT 3.95 Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is
seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach
while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
TRY IT 3.96 Jeromy can drive from his house in Cleveland to his college in Chicago in 4.5 hours. It takes his
mother 6 hours to make the same drive. Jeromy drives 20 miles per hour faster than his mother.
Find Jeromy’s speed and his mother’s speed.
In Example 3.48, the last example, we had two trains traveling the same distance. The diagram and the chart helped us
write the equation we solved. Let’s see how this works in another case.
EXAMPLE 3.49
Christopher and his parents live 115 miles apart. They met at a restaurant between their homes to celebrate his mother’s
birthday. Christopher drove 1.5 hours while his parents drove 1 hour to get to the restaurant. Christopher’s average
speed was 10 miles per hour faster than his parents’ average speed. What were the average speeds of Christopher and
of his parents as they drove to the restaurant?
Solution
Step 1. Read the problem. Make sure all the words and ideas are understood.
• Draw a diagram to illustrate what it happening. Below shows a sketch of what is happening in the example.
• Create a table to organize the information.
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• Label the columns rate, time, distance.
• List the two scenarios.
• Write in the information you know.
Step 2. Identify what we are looking for.
• We are asked to find the average speeds of Christopher and his parents.
Step 3. Name what we are looking for. Choose a variable to represent that quantity.
• Complete the chart.
• Use variable expressions to represent that quantity in each row.
• We are looking for their average speeds. Let’s let r represent the average speed of the parents. Since the
Christopher’s speed is 10 mph faster, we represent that as
Fill in the speeds into the chart.
Multiply the rate times the time to get the distance.
Step 4. Translate into an equation.
• Restate the problem in one sentence with all the important information.
• Then, translate the sentence into an equation.
• Again, we need to identify a relationship between the distances in order to write an equation. Look at the diagram
we created above and notice the relationship between the distance Christopher traveled and the distance his
parents traveled.
The distance Christopher travelled plus the distance his parents travel must add up to 115 miles. So we write:
Step 5. Solve the equation using good algebra techniques.
Now solve this equation.
So the parents' speed was 40 mph.
Table 3.12
3.5 • Solve Uniform Motion Applications 363
Christopher's speed is .
Christopher's speed was 50 mph.
Table 3.12
Step 6. Check the answer in the problem and make sure it makes sense.
Christopher drove
His parents drove
Table 3.13
Step 7. Answer the question with a complete sentence. Christopher's speed was 50 mph.
His parents' speed was 40 mph.
Table 3.14
TRY IT 3.97 Carina is driving from her home in Anaheim to Berkeley on the same day her brother is driving
from Berkeley to Anaheim, so they decide to meet for lunch along the way in Buttonwillow. The
distance from Anaheim to Berkeley is 410 miles. It takes Carina 3 hours to get to Buttonwillow,
while her brother drives 4 hours to get there. The average speed Carina’s brother drove was 15
miles per hour faster than Carina’s average speed. Find Carina’s and her brother’s average
speeds.
TRY IT 3.98 Ashley goes to college in Minneapolis, 234 miles from her home in Sioux Falls. She wants her
parents to bring her more winter clothes, so they decide to meet at a restaurant on the road
between Minneapolis and Sioux Falls. Ashley and her parents both drove 2 hours to the
restaurant. Ashley’s average speed was seven miles per hour faster than her parents’ average
speed. Find Ashley’s and her parents’ average speed.
As you read the next example, think about the relationship of the distances traveled. Which of the previous two examples
is more similar to this situation?
EXAMPLE 3.50
Two truck drivers leave a rest area on the interstate at the same time. One truck travels east and the other one travels
west. The truck traveling west travels at 70 mph and the truck traveling east has an average speed of 60 mph. How long
will they travel before they are 325 miles apart?
Solution
Step 1. Read the problem. Make sure all the words and ideas are understood.
• Draw a diagram to illustrate what it happening.
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• Create a table to organize the information.
Step 2. Identify what we are looking for.
• We are asked to find the amount of time the trucks will travel until they are 325 miles apart.
Step 3. Name what we are looking for. Choose a variable to represent that quantity.
• We are looking for the time travelled. Both trucks will travel the same amount of time. Let’s call the time t. Since
their speeds are different, they will travel different distances.
• Complete the chart.
Step 4. Translate into an equation.
• We need to find a relation between the distances in order to write an equation. Looking at the diagram, what is the
relationship between the distance each of the trucks will travel?
• The distance traveled by the truck going west plus the distance travelled by the truck going east must add up to 325
miles. So we write:
Step 5. Solve the equation using good algebra techniques.
Now solve this equation.
Table 3.15
So it will take the trucks 2.5 hours to be 325 miles apart.
Step 6. Check the answer in the problem and make sure it makes sense.
3.5 • Solve Uniform Motion Applications 365