Problem 129 - College Algebra with Modeling and Visualization

Question

A jet airliner travels 1680 miles in 3 hours with a tail wind. The return trip, into the wind, takes 3.5 hours. Find both the speed of the jet with no wind and the wind speed.

Step-by-Step Solution

Answer

The jet's speed is 520 mph, and wind speed is 40 mph.

Step 1: Define Variables

Let the speed of the jet in still air be \( j \) miles per hour and the speed of the wind be \( w \) miles per hour. We need to find these values.

Step 2: Express the Speed with Wind Effects

When traveling with the wind, the jet's speed is \( j + w \). Against the wind, the speed is \( j - w \).

Step 3: Create Equations from the Given Information

For the trip with the wind, the equation is \( \frac{1680}{j+w} = 3 \).\For the return trip against the wind, the equation is \( \frac{1680}{j-w} = 3.5 \).

Step 4: Solve the First Equation

From \( \frac{1680}{j+w} = 3 \), we get \( j + w = \frac{1680}{3} = 560 \). So, \( j + w = 560 \).

Step 5: Solve the Second Equation

From \( \frac{1680}{j-w} = 3.5 \), we get \( j - w = \frac{1680}{3.5} = 480 \). Thus, \( j - w = 480 \).

Step 6: Solve the System of Equations

We have two equations: \( j + w = 560 \) and \( j - w = 480 \). Add them: \( 2j = 1040 \), so \( j = 520 \). Subtract the second from the first: \( 2w = 80 \), so \( w = 40 \).

Step 7: Verify the Solution

Substitute \( j = 520 \) and \( w = 40 \) back into the equations: \With the wind: \( \frac{1680}{560} = 3 \), against the wind: \( \frac{1680}{480} = 3.5 \). Both satisfy the original conditions.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations that share common variables. In our problem, we have two unknowns: the speed of the jet, denoted as \( j \), and the wind speed, \( w \). We need to determine these values using the given relationships.
To resolve the problem, we make use of two equations derived from the conditions provided in the word problem:

• Equation 1: \( j + w = 560 \)
• Equation 2: \( j - w = 480 \)

Each equation represents one part of the journey, with and against the wind. Solving a system of equations like this often involves combining the equations to eliminate one variable, allowing the other variable to be easily solved. This is typically done using methods such as substitution or elimination. In this scenario, both equations are perfectly aligned for the elimination method.

Word Problems

Word problems are mathematical problems presented as a narrative, requiring interpretation to form equations. They require critical reading to identify relevant data and form equations that represent the scenario.
In our jet airliner problem, we use the context of travel to establish our equations. Identifying the journey details like distance, speeds, and times is key.
Key elements include:

• Total Distance: 1680 miles
• Time with the wind: 3 hours
• Time against the wind: 3.5 hours

We transform this data into the equations by understanding how speed, time, and distance relate. This transformation is pivotal for solving word problems, as they convert real-world scenarios into solvable mathematical equations.

Rate Problems

Rate problems involve finding relationships between distance, speed, and time. The basic formula connecting these is \( \, \text{Distance} = \text{Speed} \times \text{Time} \, \).
In this airliner problem, the rate problems become apparent in the journey times with and against the wind.

• With the wind: The speed increases, so the equation \( \, 1680 = (j + w) \times 3 \) arises.
• Against the wind: The speed decreases, leading to \( \, 1680 = (j - w) \times 3.5 \).

These equations directly derive from the concept of rate—how distance, time, and speed correlate. The rates change due to the presence of wind, affecting the overall speed during different parts of the journey.

Linear Equations

Linear equations are algebraic equations where each term is a constant or the product of a constant and a single variable. In our exercise, the linear equations used are simple and involve basic operations.
The system provided:

• \( j + w = 560 \)
• \( j - w = 480 \)

These equations only involve addition, subtraction, and multiplication, making them linear. Solving them benefits from understanding how changes in one variable influence the other through direct relationships.
Linear systems of this type can often be quickly visualized as lines on a coordinate plane where the solution represents the intersection point, if any, signaling the values of the variables that satisfy both equations simultaneously.