Problem 53 - Algebra for College Students

Question

Two airplanes leave Chicago at the same time and fly in opposite directions. If one travels at 450 miles per hour ind the other at 550 miles per hour, how long will it ake for them to be 4000 miles apart?

Step-by-Step Solution

Answer

It will take 4 hours for the airplanes to be 4000 miles apart.

Step 1: Understand the Problem

We have two airplanes starting at the same point and moving in opposite directions. We need to calculate how long it will take for the airplanes to be 4000 miles apart. One airplane travels at 450 miles per hour and the other at 550 miles per hour.

Step 2: Calculate Relative Speed

Since the airplanes are moving in opposite directions, we add their speeds to find the rate at which they are separating from each other. Thus, the relative speed is 450 + 550 = 1000 miles per hour.

Step 3: Set Up the Equation

We set up the equation using the formula: \( \text{Distance} = \text{Rate} \times \text{Time} \). Here, the distance is 4000 miles and the rate is the relative speed of 1000 miles per hour. The equation is: 4000 = 1000 \times t, where \( t \) represents the time in hours.

Step 4: Solve for Time

To find \( t \), we need to isolate it in the equation. Divide both sides of the equation by 1000: \( t = \frac{4000}{1000} \). Therefore, \( t = 4 \) hours.

Key Concepts

Distance-Rate-Time Relationship

Let's dive into the distance-rate-time relationship, a fundamental concept in solving motion problems. This relationship stems from the formula:

• \( \text{Distance} = \text{Rate} \times \text{Time} \)

In this formula:

• "Distance" refers to how far an object travels.
• "Rate" denotes the speed at which the object moves.
• "Time" represents the duration over which the travel occurs.

This principle helps us understand how the three variables interact. If we know any two of these, we can solve for the third.
For instance, if an airplane flies 300 miles at 150 miles per hour, the time spent flying is calculated by rearranging the equation to solve for time:

• \( \text{Time} = \frac{\text{Distance}}{\text{Rate}} \)

Getting comfortable with manipulating this formula is key in making sense of many algebra word problems.

Solving Algebraic Equations

Understanding how to solve algebraic equations is crucial for tackling math problems that involve unknown variables. In algebra, equations represent a balance or equality between two expressions.
To solve for an unknown, follow these steps:

• Identify the variable you need to solve for – this is your unknown quantity.
• Use operations to isolate the variable on one side of the equation.
• Simplify the equation by performing the same operation on both sides.

For example, if you have the equation \( 4000 = 1000 \times t \), to isolate \( t \), you'll divide both sides by 1000:

• \( t = \frac{4000}{1000} \)
• \( t = 4 \)

This tells us that the time \( t \) is 4 hours. Successfully solving algebraic equations requires practice and a clear understanding of maintaining balance in the equation.

Word Problems in Algebra

Word problems in algebra may seem daunting, but they are merely real-world situations translated into mathematical expressions. The key is to break down the scenario step by step.
Start by identifying what you know and what you need to find:

• First, understand the problem entirely. Recognize the key information given, such as speeds or distances.
• The next step is to outline a strategy: what relationship or formula will help? Use the distance-rate-time formula, for example, when dealing with motion problems.
• Translate the word problem into an algebraic equation.

After setting up the equation, solve, and interpret the solution back in terms of the original problem:

• Ask yourself: does this solution make sense given the context?

Remember, practice is vital for mastering word problems. With repeated exposure, converting a real-world problem into an algebraic solution becomes intuitive.