Problem 103 - College Algebra with Modeling and Visualization

Question

Motion A car went 372 miles in 6 hours, traveling part of the time at 55 miles per hour and part of the time at 70 miles per hour. How long did the car travel at each speed?

Step-by-Step Solution

Answer

3.2 hours at 55 mph, 2.8 hours at 70 mph.

Step 1: Define Variables

Let \( t_1 \) be the time (in hours) the car traveled at 55 mph, and \( t_2 \) be the time (in hours) the car traveled at 70 mph.

Step 2: Set up Equations

We have two conditions: the total travel time and the total distance. The total time is 6 hours: \( t_1 + t_2 = 6 \). The total distance is 372 miles: \( 55t_1 + 70t_2 = 372 \).

Step 3: Solve the System of Equations (Equation 1)

From the equation \( t_1 + t_2 = 6 \), express \( t_1 \) in terms of \( t_2 \): \( t_1 = 6 - t_2 \).

Step 4: Substitute and Solve for \( t_2 \) (Equation 2)

Substitute \( t_1 = 6 - t_2 \) into the distance equation: \( 55(6-t_2) + 70t_2 = 372 \). Expand and simplify to get: \( 330 - 55t_2 + 70t_2 = 372 \). This simplifies to \( 15t_2 = 42 \). Thus, \( t_2 = \frac{42}{15} = 2.8 \).

Step 5: Solve for \( t_1 \)

Using \( t_1 = 6 - t_2 \), substitute \( t_2 = 2.8 \) to find \( t_1: t_1 = 6 - 2.8 = 3.2 \).

Step 6: Conclusion

The car traveled for 3.2 hours at 55 mph and 2.8 hours at 70 mph.

Key Concepts

Motion Problems

Motion problems often involve questions about objects like cars, trains, or planes moving at various speeds over time. These problems require us to determine how long they travel at different speeds, how far they go, or at what rate they are moving.

In a typical motion problem, like the one in the original exercise, we need to consider different parts of a journey. The car travels at two different speeds and we need to find out the time spent traveling at each speed. This can seem complex at first, but by breaking it down into understandable parts, it becomes more manageable.

• Think about what is changing during the journey: speed, time, and distance.
• Understand that motion problems are often solved by setting up equations that link these variables.
• Use logical reasoning to define variables, create equations, and solve them.

These tools can help simplify any motion problem, making it easier to understand and solve.

Distance-Rate-Time Relationship

The distance-rate-time relationship is a fundamental concept in motion problems. It defines how distance traveled is directly related to the speed (or rate) and the time of travel. This can be written as the formula: \[ Distance = Rate imes Time\] In the context of the problem at hand, we use this relationship repeatedly.

For the car, we have one equation for each part of the journey:

• When traveling at 55 mph, the distance covered is \( 55t_1 \).
• When traveling at 70 mph, the distance covered is \( 70t_2 \).

These components together add up to the total distance, 372 miles. The strength of the distance-rate-time formula is its versatility. It allows the conversion of speeds and times into a tangible measurement of distance, simplifying complex motion problems into manageable calculations.

Linear Equations

Linear equations are a crucial tool in solving motion problems, especially when dealing with multiple scenarios and conditions. A linear equation is an equation between two variables that gives us a straight line when graphed. This is handy in motion problems, where we deal with ideas like total distance and time, which are interrelated linearly.

In the original exercise, we use linear equations to express two key conditions:

• Total time equation: \( t_1 + t_2 = 6 \)
• Total distance equation: \( 55t_1 + 70t_2 = 372 \)

These equations are solved simultaneously to find the unknown variables \( t_1 \) and \( t_2 \), which represent time spent at each speed. By expressing one variable in terms of another (e.g., \( t_1 = 6 - t_2 \)), and substituting back, we simplify the equations. This makes finding a solution much more straightforward. Understanding linear equations is invaluable for analyzing and solving motion-based questions.