Problem 68 - College Algebra with Corequisite Support

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. A jeep and BMW enter a highway running east-west at the same exit heading in opposite directions. The jeep entered the highway 30 minutes before the BMW did, and traveled 7 mph slower than the BMW. After 2 hours from the time the BMW entered the highway, the cars were 306.5 miles apart. Find the speed of each car, assuming they were driven on cruise control.

Step-by-Step Solution

Answer

BMW speed: 72 mph; Jeep speed: 65 mph.

Step 1: Define the variables

Let the speed of the BMW be \( b \) miles per hour. Then the speed of the Jeep is \( b - 7 \) miles per hour, since it's 7 mph slower than the BMW.

Step 2: Write the equation for the BMW's distance

The BMW traveled for 2 hours, so the distance it covers is given by \( 2b \).

Step 3: Write the equation for the Jeep's distance

The Jeep travels for 2.5 hours (since it entered 30 minutes earlier than the BMW). Thus, the distance the Jeep covers is \( (b-7) \times 2.5 \).

Step 4: Set up the equation based on total distance

The sum of the distances covered by both cars is 306.5 miles. Therefore, the equation is \( 2b + 2.5(b-7) = 306.5 \).

Step 5: Solve the equation

Expand and simplify the equation: \( 2b + 2.5b - 17.5 = 306.5 \). Combine like terms to get \( 4.5b - 17.5 = 306.5 \). Add 17.5 to both sides: \( 4.5b = 324 \). Divide both sides by 4.5 to find \( b = 72 \).

Step 6: Calculate the speed of the Jeep

Since the speed of the BMW is 72 mph, and the Jeep is 7 mph slower, the speed of the Jeep is \( 72 - 7 = 65 \) mph.

Key Concepts

Systems of Equations

When solving a problem involving multiple unknowns, such as finding the speed of two different cars, we use systems of equations. A system of equations is a set of two or more equations that have common variables. In the exercise, we are given that the jeep and BMW have speeds that relate to each other. The BMW's speed is one variable, denoted as \( b \), and the jeep's speed is \( b - 7 \), since it is 7 mph slower.

• We establish equations based on these relationships.
• We use the cumulative distance both vehicles traveled to frame these equations.
• The goal is to find values for \( b \) and the jeep's speed simultaneously.

By setting up a system of equations, we harness the power of algebraic methods to uncover the numerical values of these speeds efficiently.

Distance Problems

Distance problems like this one involve calculating how far an object travels based on specific conditions. The essential formula for distance is \( \text{distance} = \text{speed} \times \text{time} \). In this exercise, two cars travel at different speeds and times, but after a while, they are a known distance apart.

• The BMW travels for 2 hours, while the jeep travels for 2.5 hours, as it started 30 minutes earlier.
• We use the individual distances driven, along with the total separation distance, to construct our main equation.

Understanding the roles of speed and time allows us to link the parts of this problem and solve for the unknown variables. This understanding is crucial in many real-world scenarios where travel details need to be computed quickly and accurately.

Speed Calculation

Speed calculation in this problem is central to finding the solution. Once the system of equations is in place, we compute speed using algebra. Speed is generally expressed in units like miles per hour, and is the rate of motion.

• For the BMW, its speed is a straightforward variable: \( b \).
• For the jeep, given that it moves slower, its speed is \( b - 7 \).

The equation formed from the total distance lets us calculate \( b \). Solving this gives us the BMW's speed, which in turn helps us derive the jeep's speed:- Calculate \( b \) using the final equation: \( 4.5b = 324 \).- Find \( b = 72 \) for the BMW.- Subtract 7 to find the jeep's speed, \( 65 \), which confirms the calculations.

Algebra Problem Solving

Algebra problem solving is about breaking down a problem into manageable pieces and using logical steps to find a solution. In our exercise, algebra is key to understanding how each detail, like car speed and travel time, interacts to form a coherent picture of the journey.
Here are some steps we follow:

• Identify the variables and how they relate to the problem.
• Set up equations based on the relationships and conditions provided.
• Solve these equations step by step, checking at each stage to ensure accuracy.
• Reinterpret the solution back into the context of the problem to validate it.

Algebra is a versatile tool not only for solving for unknowns but also for making sense of complex, multi-faceted problems encountered in academics and daily life.