Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem analytically, and support your solution graphically. 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

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

Step 1: Define Variables

Let the time spent traveling at 55 mph be denoted as \( t_1 \) and the time spent at 70 mph be denoted as \( t_2 \).

Step 2: Sum of Times Equation

The total travel time is given as 6 hours. Therefore, we can write the first equation: \( t_1 + t_2 = 6 \).

Step 3: Distance Equation Using Speeds

The car traveled a total of 372 miles. The distance covered at 55 mph is \( 55t_1 \) and at 70 mph is \( 70t_2 \). The second equation is: \( 55t_1 + 70t_2 = 372 \).

Step 4: Substitute and Solve for Time Variables

From Step 2, express \( t_1 \) as \( t_1 = 6 - t_2 \) and substitute into the second equation: \( 55(6 - t_2) + 70t_2 = 372 \). This simplifies to \( 330 - 55t_2 + 70t_2 = 372 \), leading to \( 15t_2 = 42 \). Thus, \( t_2 = \frac{42}{15} = 2.8 \) hours.

Step 5: Solve for the Other Time Variable

Substitute \( t_2 = 2.8 \) back into \( t_1 = 6 - t_2 \) to find \( t_1 \): \( t_1 = 6 - 2.8 = 3.2 \) hours.

Step 6: Graphical Representation

Plot the equation \( 55t_1 + 70t_2 = 372 \) on a coordinate plane with \( t_1 \) on the x-axis and \( t_2 \) on the y-axis. The intersection with \( t_1 + t_2 = 6 \) will confirm our solution of \( t_1 = 3.2 \) and \( t_2 = 2.8 \).

Key Concepts

Linear Equations

Linear equations are a foundational concept in algebra used to find an unknown variable when you have enough information to describe a linear relationship.
They are characterized by variables raised to the power of one, and they result in a graph that takes the shape of a straight line.
In our problem, we used linear equations to represent the total time and distance covered by the car as it traveled at two different speeds:

• The first equation, \( t_1 + t_2 = 6 \), shows that the sum of the hours spent driving must equal 6.
• The second equation, \( 55t_1 + 70t_2 = 372 \), represents the total distance of 372 miles driven at the two different speeds.

Linear equations help us systematically organize the information we have and calculate the unknown values accurately.

Substitution Method

The substitution method is a technique used to solve systems of equations where one of the equations is solved for one variable, which is then substituted into another equation.
This method simplifies the process by reducing the number of variables in the equation we work on first.
In this problem, we started by expressing \( t_1 \) from the first equation:

• From \( t_1 + t_2 = 6 \), we derived \( t_1 = 6 - t_2 \).
• This expression for \( t_1 \) was then substituted into the second equation: \( 55(6 - t_2) + 70t_2 = 372 \).

After substitution, we had a single equation with one variable \( t_2 \), making it easier to solve for \( t_2 \).
Once we found \( t_2 \), we could substitute it back to find \( t_1 \).

Graphical Solution

Graphical solutions provide a visual approach to problem-solving by representing equations on a graph. This enables us to see where equations intersect, giving us the solution to the problem graphically.
For our exercise, both equations are plotted on the same axes:

• The line \( t_1 + t_2 = 6 \) appears, where \( t_1 \) is on the x-axis and \( t_2 \) on the y-axis.
• Another line for \( 55t_1 + 70t_2 = 372 \) intersects with the first, at the point that provides the solution \( t_1 = 3.2 \) and \( t_2 = 2.8 \).

Graphical methods are particularly useful when you want to verify the accuracy of an algebraic solution or understand the relationship between quantitative variables.

Distance-Time Problems

Distance-time problems are mathematical challenges that involve calculating the relationship between the distance traveled by an object and the time it takes, often factoring in different speeds.
These problems often require setting up equations based on the formula \( \Distance = \Speed \times \Time \) and solving for the unknowns.
In our exercise:

• The car travels a known distance of 372 miles across two different speeds: 55 mph and 70 mph.
• We created equations to express how much time \( t_1 \) was spent at 55 mph and \( t_2 \) was spent at 70 mph, totaling 6 hours.

By understanding the relationship between distance, time, and speed, and setting up our equations correctly, we calculated the time spent at each speed efficiently.