Problem 59 - Algebra and Trigonometry

Question

Distance, Speed, and Time A pilot flew a jet from Montreal to Los Angeles, a distance of 2500 \(\mathrm{mi}\) . On the return trip, the average speed was 20\(\%\) faster than the outbound speed. The round-trip took 9 \(\mathrm{h} 10 \mathrm{min}\) . What was the speed from Montreal to Los Angeles?

Step-by-Step Solution

Answer

The speed from Montreal to Los Angeles was approximately 272.73 mi/h.

Step 1: Define variables

Let the speed from Montreal to Los Angeles be \( x \) (in miles per hour). Then the speed on the return trip from Los Angeles to Montreal is \( 1.2x \) (20\% faster).

Step 2: Set up time equations

The time taken to travel from Montreal to Los Angeles is \( \frac{2500}{x} \) hours, and the time taken for the return trip is \( \frac{2500}{1.2x} \) hours.

Step 3: Convert total trip time

The total round trip time is 9 hours and 10 minutes. Convert this to hours: 9 hours and 10 minutes = 9 + \( \frac{10}{60} \) = 9.1667 hours.

Step 4: Formulate the equation

Set up the equation for the total trip time: \[ \frac{2500}{x} + \frac{2500}{1.2x} = 9.1667 \]

Step 5: Solve for x

Multiply through by \( 1.2x \) to clear the denominators: \[ 2500(1.2) + 2500 = 9.1667(1.2x) \] which simplifies to \[ 2500x + 3000x = 11x \times 9.1667 \] then solve for \( x \).

Step 6: Further simplification

Combine the terms: \( 5500 = 11 \times 9.1667x \). Simplify and solve for \( x \): \[ x = \frac{5500}{11 \times 9.1667} \approx 272.73 \]\( \text{mi/h} \).

Key Concepts

Distance Formula

The distance formula is a key component in solving problems involving speed, distance, and time. It is expressed as:

• Distance = Speed × Time.

This formula helps us determine how far one can travel at a certain speed over a given period.
In our problem, we use the formula to set up equations for the time taken on each segment of the journey.
From Montreal to Los Angeles, the pilot travels a distance of 2500 miles at an unknown speed, so the time taken is \( \frac{2500}{x} \) hours. On the return trip, where the speed is 20% faster, the time is \( \frac{2500}{1.2x} \) hours.
These equations allow us to determine the unknown variable, speed.

Algebraic Equation

An algebraic equation is an expression that establishes equality between two expressions. It contains one or more variables.
In this exercise, we use an algebraic equation to find the unknown speed of the jet.
After defining the speeds for both directions of the journey, we set up the equation:

• \( \frac{2500}{x} + \frac{2500}{1.2x} = 9.1667 \)

Here, \(x\) represents the speed from Montreal to Los Angeles, and we need to solve this equation to find its value.
We then use algebraic manipulations, such as clearing the denominators and combining terms, to isolate \(x\) and find its value.

Variable Definition

Defining variables is crucial in understanding and solving mathematical problems.
Variables are symbols, often letters, that represent unknown values in equations. In this problem:

• We define the speed from Montreal to Los Angeles as \( x \),
• The speed on the return flight is \( 1.2x \), as it is 20\% faster.

These definitions allow us to construct equations that accurately represent the relationships within the problem.
By clearly defining variables before forming equations, it simplifies the problem-solving process.

Proportional Increase

A proportional increase involves scaling a quantity by a given percentage to determine a new value.
This concept is crucial in our problem when calculating the return speed of the jet.
The return speed is described as 20% faster than the outbound speed.

• To express this increase mathematically, we multiply the original speed \( x \) by 1.2.

This factor (1.2) reflects the new speed as it is 120% of the original, accounting for the increase.
Understanding proportional increase helps in constructing more accurate models and predictions based on percentage changes.