Problem 42 - Intermediate Algebra

Question

An airplane leaves Los Angeles bound for Caracas, Venezuela, flying at an average rate of 500 mph. At the same time, another airplane leaves Caracas bound for Los Angeles, averaging 550 mph. If the airports are \(3,675\) miles apart, when will the air traffic controllers have to make the pilots aware that the planes are passing each other?

Step-by-Step Solution

Answer

The planes pass each other after 3.5 hours.

Step 1: Understand the Problem

The exercise asks us to find the time at which two airplanes passing each other. We know that one plane leaves Los Angeles for Caracas at 500 mph, and another leaves Caracas for Los Angeles at 550 mph. The distance between Los Angeles and Caracas is 3,675 miles.

Step 2: Set Up the Equation

Let the time it takes for the planes to meet be denoted by \( t \) hours. The distance covered by the first plane in \( t \) hours is \( 500t \) miles, and the distance covered by the second plane is \( 550t \) miles. Together, they need to cover the total distance of 3,675 miles. Therefore, we set up the equation: \[ 500t + 550t = 3675 \]

Step 3: Simplify the Equation

Combine the terms in the equation: \[ 500t + 550t = 1050t \] Thus, the equation becomes: \[ 1050t = 3675 \]

Step 4: Solve for Time \( t \)

To solve for \( t \), divide both sides of the equation by 1,050: \[ t = \frac{3675}{1050} \]

Step 5: Calculate the Exact Value of \( t \)

Perform the division to find \( t \): \[ t = \frac{3675}{1050} = 3.5 \] This tells us that it takes 3.5 hours for the two planes to pass each other.

Key Concepts

Rate and Distance Problems

In rate and distance problems, we explore the relationship between speed, distance, and time. It is essential to remember the fundamental formula:

• Distance = Rate × Time.

This equation is the backbone of these problems. When you know two of the three elements—distance, rate, or time—you can find the third.
In our airplane scenario, the challenge is to determine when they pass each other. Here, the combined rate is the sum of their speeds because both planes are moving towards each other. Hence, we use:

• Combined rate = 500 mph + 550 mph = 1050 mph.

This combined rate allows us to treat the problem as if one plane was traveling the total distance by itself. This simplifies the finding of when they meet by setting the product of their combined speed and time equal to the total distance. Understanding these principles helps in tackling more complex real-world scenarios.

Simultaneous Equations

Simultaneous equations are tools that solve problems where two or more equations represent a scenario. In our problem, although only one equation is visible, simultaneous thinking helps in conceptualizing how both planes move simultaneously towards each other.
The equation we set,

• \(500t + 550t = 3675\),

actually encapsulates two distances: one covered by each plane. Each airplane's movement contributes to reaching a common point, hence leading us to solve both from a shared perspective.
Using simultaneous equations often requires working with two equations at once. But here, we've smartly combined them into a single equation by adding the distances the two planes cover. This is a simple yet effective use of simultaneous equation strategies without overcomplicating the scenario.
By consolidating into a single equation, we can smoothly solve for the one variable, time \(t\), needed for their meeting.

Algebraic Reasoning

Algebraic reasoning refers to the manipulation of variables and equations to solve problems. In our plane example, algebraic reasoning helps transform verbal information into a solvable mathematical model.
Key steps involve:

• Identifying variables: In this problem, \(t\) represents time.
• Setting up equations: Use known values to create equations, like \(500t + 550t = 3675\).
• Simplifying expressions: Combine like terms, e.g., \(500t + 550t = 1050t\).
• Solving the equation: Apply mathematical operations to isolate and find the value of \(t\).

Algebraic reasoning is not just about crunching numbers; it involves thoughtful understanding of how parts of the problem relate, allowing for strategic substitutions and simplifications.
This demonstrates how algebra serves as a powerful tool in decoding and solving real-world problems, and it's a skill that can make complex situations much more manageable.