Problem 55 - Prealgebra and Introductory Algebra

Question

A commercial jet can fly 550 mph in calm air. Traveling with the jet stream, the plane can fly \(2400 \mathrm{mi}\) in the same amount of time it takes to fly \(2000 \mathrm{mi}\) against the jet stream. Find the rate of the jet stream.

Step-by-Step Solution

Answer

The rate of the jet stream is \(100\) mph.

Step 1: Define the Variables

Let's denote the speed of the jet as 'j', and the speed of the plane in calm air as 'p'. According to the problem, p is 550 mph and j is unknown.

Step 2: Set Up the Equations

The time it takes to fly a certain distance is equal to the distance divided by the speed. So, we can make two equations from the problem: \n1) For the trip with the Jetstream the equation is \(2400 / (p + j) = t\)\n2) For the trip against the Jetstream the equation is \(2000 / (p - j) = t\)

Step 3: Solve the System of Equations

The problem states that the times are equal, so we can set the equations equal to each other. This gives us \(2400 / (p + j) = 2000 / (p - j)\). Cross multiply to obtain \(2400(p - j) = 2000(p + j)\). After simplifying, this gives \(480,000 - 2400j = 2000p + 2000j\). Substituting p=550 into the equation gives \(480000 - 2400j = 1100000 + 2000j\). Solving this equation gives \(j = 100\) mph.

Step 4: Check the Result

Substitute j=100 into the original equations to validate this result.

Key Concepts

Jet Stream

Flying with or against the jet stream can significantly impact a plane's travel time and speed. A jet stream is a fast-flowing, narrow air current found in the atmosphere. Jet streams often flow from west to east and can reach speeds of up to 200 mph. When an airplane flies in the same direction as the jet stream, it effectively receives a boost in speed, reducing its travel time. Conversely, flying against the jet stream means that the plane faces a headwind, slowing it down. In problems involving jet streams, we must account for this additional speed factor when determining overall travel times.

Equations

Understanding how to set up equations is crucial in solving rate problems. In the given exercise, we're asked to find the rate of the jet stream using distances and known speeds. The step-by-step approach creates equations based on distance and speed relationships. For a trip with the jet stream, the equation is:

• \(\frac{2400}{p + j} = t\)

against the jet stream, it becomes:

• \(\frac{2000}{p - j} = t\)

Here, \(t\) represents the time variable, \(p\) the plane speed in calm air, and \(j\) the jet stream speed. These equations model the relationship between distance, speed, and time, providing a foundation for deriving the jet stream speed.

Distance-Time Relationship

In rate problems, the relationship between distance, speed, and time is a central concept. The basic formula is:

• Distance = Speed × Time

In this context, knowing two of these variables allows you to calculate the third. For example, if the distance is known (e.g., 2400 miles), and the speed can be expressed in terms of the jet stream (\(p + j\) or \(p - j\)), you can find the time. Then, equating travel times for both directions (with and against the jet stream) enables us to solve for the unknown variable matching this relationship.

Speed Calculation

Calculating speed involves solving the equations for the jet stream rate, leveraging the distance-time relationship. In the example, equating both times gives us:

• \(\frac{2400}{(550 + j)} = \frac{2000}{(550 - j)}\)

Cross-multiplying simplifies the equation to include only the variable \(j\). The resulting equation is solved through algebraic manipulation:

• \(2400(550 - j) = 2000(550 + j)\)
• Simplify to \(480,000 - 2400j = 1,100,000 + 2000j\)
• Solving yields \(j = 100\) mph

This confirms the jet stream speed is 100 mph, illustrating the interaction of speed, time, and distance in calculation.