Problem 17 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Two airplanes leave an airport at the same time, flying in opposite directions. One plane is flying at twice the speed of the other. If after 4 hours they are \(1,800\) miles apart, find the speed of each plane.

Step-by-Step Solution

Answer

The speeds are 150 mph for the slower plane and 300 mph for the faster plane.

Step 1 - Define Variables

Let the speed of the slower airplane be denoted as \(s\). Since the second airplane is flying at twice the speed, its speed will be \(2s\).

Step 2 - Write Distance Formulas

The distance traveled by an airplane is given by the product of its speed and time. In 4 hours, the slower airplane covers a distance of \(4s\), and the faster airplane covers a distance of \(4 \cdot 2s = 8s\).

Step 3 - Set Up the Equation

Since the airplanes are flying in opposite directions, the sum of their distances after 4 hours equals the total distance between them, which is 1,800 miles. Therefore, the equation is: \(4s + 8s = 1,800\).

Step 4 - Solve for s

Combine like terms to get: \(12s = 1,800\). To find the value of \(s\), divide both sides by 12: \(s = \frac{1,800}{12} = 150\).

Step 5 - Find the Speed of the Faster Plane

Since the faster airplane's speed is twice that of the slower one, the speed is: \(2s = 2 \cdot 150 = 300\).

Step 6 - Conclude with Speeds

Thus, the speed of the slower airplane is 150 miles per hour and the speed of the faster airplane is 300 miles per hour.

Key Concepts

Defining Variables

When solving verbal problems algebraically, it's crucial to start by defining your variables.
Variables represent unknown values and help in setting up equations.
In the given problem, we want to find the speeds of two airplanes.
To do this:

• Let's call the speed of the slower airplane \(s\).
• Since the faster airplane is flying at twice the speed of the slower one, its speed is \(2s\).

Defining variables simplifies your calculations and keeps your equations clear and organized.
Always assign a meaningful letter to each variable to avoid confusion later.

Distance Formula

The Distance Formula is fundamental in problems involving movement over time.
The formula is:

• Distance = Speed \( \times \) Time

In our problem, both planes travel for 4 hours in opposite directions.
Let’s use the distances each airplane travels:

• The slower airplane's distance: \(4s\)
• The faster airplane's distance: \(4 \times 2s = 8s\)

These are the expressions for their distances after 4 hours, based on their speeds and the time they've been flying.
The Distance Formula helps us set up accurate equations to find unknown values like speed or distance.

Setting Up Equations

Setting up equations correctly is key to solving algebraic problems.
In this problem, the sum of the distances the airplanes travel must equal the total distance between them after 4 hours, which is 1,800 miles.
So, we set up the equation: \[4s + 8s = 1,800\] This equation represents the total distance.
Next, combine like terms: \[12s = 1,800\] Solving for \(s\) involves dividing both sides by 12: \[s = \frac{1,800}{12} = 150\] Therefore, the speeds are:

• Slower airplane: 150 miles per hour
• Faster airplane: 300 miles per hour \(2 \times 150\)

To solve any algebraic problem, ensure you write out your equations clearly and isolate the unknown variable.
This guarantees you'll find the correct solution.