MATH 125 Section 3.5 - Pasadena City College | MATH125 Section 3.5 - Pasadena City College

MATH 125 Section 3.5 - Pasadena City College | MATH125 Section 3.5 - Pasadena City College Solve Uniform Motion Applications Learning Objectives By the end of this section, you will be able to: Solve uniform motion applications Be Prepared! Before you get started, take this readiness quiz. 1. Find the distance travelled by a car going 70 miles per hour for 3 hours. If you missed this problem, review Example 2.58. 2. Solve x + 1.2(x − 10) = 98. If you missed this problem, review Example 2.39. 3. Convert 90 minutes to hours. If you missed this problem, review Example 1.140. Solve Uniform Motion Applications When planning a road trip, it often helps to know how long it will take to reach the destination or how far to travel each day. We would use the distance, rate, and time formula, D = rt, which we have already seen. In this section, we will use this formula in situations that require a little more algebra to solve than the ones we saw earlier. Generally, we will be looking at comparing two scenarios, such as two vehicles travelling at different rates or in opposite directions. When the speed of each vehicle is constant, we call applications like this uniform motion problems. Our problem-solving strategies will still apply here, but we will add to the first step. The first step will include drawing a diagram that shows what is happening in the example. Drawing the diagram helps us understand what is happening so that we will write an appropriate equation. Then we will make a table to organize the information, like we did for the money applications. The steps are listed on the next page for easy reference: Chapter 3 Math Models 369EXAMPLE 3.48 An express train and a local train leave Pittsburgh to travel to Washington, D.C. The express train can make the trip in 4 hours and the local train takes 5 hours for the trip. The speed of the express train is 12 miles per hour faster than the speed of the local train. Find the speed of both trains. Solution Step 1. Read the problem. Make sure all the words and ideas are understood. • Draw a diagram to illustrate what it happening. Shown below is a sketch of what is happening in the example. • Create a table to organize the information. • Label the columns “Rate,” “Time,” and “Distance.” HOW TO : : USE A PROBLEM-SOLVING STRATEGY IN DISTANCE, RATE, AND TIME APPLICATIONS. Read the problem. Make sure all the words and ideas are understood. ◦ Draw a diagram to illustrate what it happening. ◦ Create a table to organize the information. ◦ Label the columns rate, time, distance. ◦ List the two scenarios. ◦ Write in the information you know. Identify what we are looking for. 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. ◦ Multiply the rate times the time to get the distance. Translate into an equation. ◦ Restate the problem in one sentence with all the important information. ◦ Then, translate the sentence into an equation. Solve the equation using good algebra techniques. Check the answer in the problem and make sure it makes sense. Answer the question with a complete sentence. Step 1. Step 2. Step 3. Step 4. Step 5. Step 6. Step 7. 370 Chapter 3 Math Models This OpenStax book is available for free at • 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 r + 12. r = speed of the local train r + 12 = speed of the express train 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: 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. Chapter 3 Math Models 371express train 60 mph (4 hours) = 240 miles local train 48 mph (5 hours) = 240 miles 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. • 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 r + 10. Fill in the speeds into the chart. 372 Chapter 3 Math Models This OpenStax book is available for free at - - - - - - - - -- Continued