Problem 73 - Precalculus: Mathematics for Calculus

Question

Distance, Speed, and Time John and Mary leave their house at the same time and drive in opposite directions. John drives at \(60 \mathrm{mi} / \mathrm{h}\) and travels \(35 \mathrm{mi}\) farther than Mary, who drives at \(40 \mathrm{mi} / \mathrm{h}\). Mary's trip takes 15 min longer than John's. For what length of time does each of them drive?

Step-by-Step Solution

Answer

John drives for 2.25 hours and Mary drives for 2.5 hours.

Step 1: Define Variables

Let's define the variables for John's and Mary's travel time. Let \( t \) be the time, in hours, that John drives. Hence, Mary's driving time is \( t + \frac{1}{4} \) because her trip is 15 minutes (which is \( \frac{1}{4} \) hours) longer than John's.

Step 2: Write Equations for Distance

John's driving distance can be expressed as \( 60t \). Mary's driving distance is \( 40(t + \frac{1}{4}) \). We are given that John travels 35 miles farther than Mary. Thus, we can write the equation: \[ 60t = 40(t + \frac{1}{4}) + 35 \]

Step 3: Simplify Mary's Distance Equation

Distribute the \( 40 \) in Mary's distance expression: \[ 40(t + \frac{1}{4}) = 40t + 10 \]

Step 4: Solve for John's Time

Substitute the simplified expression for Mary's distance into the original equation and solve for \( t \): \[ 60t = 40t + 10 + 35 \]\[ 60t = 40t + 45 \]Subtract \( 40t \) from both sides:\[ 20t = 45 \]Divide both sides by 20:\[ t = \frac{45}{20} = 2.25 \]

Step 5: Determine Mary's Time

Use the equation \( t + \frac{1}{4} \) for Mary's travel time and substitute \( t = 2.25 \):\[ 2.25 + \frac{1}{4} = 2.25 + 0.25 = 2.5 \]

Step 6: Interpret the Result

John drives for 2.25 hours, and Mary drives for 2.5 hours.

Key Concepts

Distance-Rate-Time Relationship

Understanding the relationship between distance, rate (or speed), and time is crucial for solving many math problems, especially when dealing with travel scenarios. To begin, this relationship is often expressed with the simple equation:

• Distance = Rate × Time

This formula tells us that the distance traveled is equal to the speed of travel multiplied by the time spent traveling. When you know any two of these variables, you can easily solve for the third. For example:

• If you know the distance and rate, divide the distance by the rate to find time.
• If you know the distance and time, divide the distance by time to find the rate.

In our exercise, John and Mary's travel involves John driving at 60 miles per hour, and Mary at 40 miles per hour. Using our formula, we can express their distances in terms of time, allowing us to set up equations to solve the problem.

Word Problems

Word problems like the one involving John and Mary are common in math courses because they apply mathematical concepts to real-world scenarios. Here are a few tips to tackle word problems effectively:

• Read Carefully: Understand the details and conditions presented in the problem. Identify what you are solving for.
• Translate Into Mathematics: Convert the word problem into equations using known formulas or relationships. Define your variables clearly.
• Identify Clues: Look for hints that provide pivotal information, like time differences (e.g., Mary's trip takes longer) or relative distances (e.g., John travels 35 miles farther).

By clearly defining the variables, and understanding the relationships and conditions, you can set up equations that represent the scenario accurately. This understanding is key to finding a solution.

Solving Equations

After setting up the equations based on the word problem, the next step is to solve them. Let's break down how this is done through the example problem:1. **Substitute and Simplify:** - We establish equations using the relationship between distance, rate, and time. For instance, John's distance is expressed as \(60t\) and Mary's as \(40(t + \frac{1}{4})\). - Simplify these equations where possible. Note that simplifying helps avoid confusion and errors down the line.
2. **Isolate the Variable:** - Rearrange the equations to isolate the variable. In our problem, subtract 40t from both sides to focus on solving for the time variable \(t\).
3. **Solve and Verify:** - Once \(t\) was isolated, we divided both sides by 20 to find \(t\), which solves John's driving time. - Always check back with the conditions in the word problem to verify that your solutions make sense. For example, adding the 15 minutes to John's time to get Mary's duration ensures completeness.
Solving equations requires practice but becomes more intuitive through understanding and applying basic algebraic principles. This will be handy not only in math tests but also in real-life problem-solving scenarios.