Problem 46 - Algebra 1: Concepts and Skills

Question

In Exercises 46–48, use the following information. A boat moves through still water at x kilometers per hour (km/h). It travels 24 km upstream against a current of 2 km/h and then returns to the starting point with the current. The rate upstream is x-2 because the boat moves against the current, and the rate downstream is x 2 because the boat moves with the current. Write an algebraic model for the total time for the round trip.

Step-by-Step Solution

Answer

The algebraic model for the total time for the round trip of the boat is \( \text{Total Time} = \frac{24}{x-2} + \frac{24}{x+2} \) hours.

Step 1: Calculate Time Upstream

Time taken to travel upstream is calculated by \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}.\) Here, the speed is \(x-2\) km/h and distance is 24 km. So, the time upstream \(\text{Time}_\text{up} = \frac{24}{x-2}\) hours.

Step 2: Calculate Time Downstream

Similarly, time taken to travel downstream is also calculated by \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\). Here, the speed is \(x+2\) km/h and distance is same 24 km. So, the time downstream \(\text{Time}_\text{down} = \frac{24}{x+2}\) hours.

Step 3: Algebraic Model for Total Time

The total time for the round trip is the sum of the time taken to go upstream and the time taken to go downstream. So, using the results from step 1 and step 2, our algebraic model becomes: \(\text{Total Time} = \text{Time}_\text{up} + \text{Time}_\text{down} = \frac{24}{x-2} + \frac{24}{x+2}\) hours.

Key Concepts

Speed and Distance Problems

Speed and distance problems involve calculating how long it takes to travel a certain distance at a given speed. These problems usually have a real-world context, such as traveling by car, boat, or walking.
For example, in our exercise, the boat needs to travel upstream and downstream over the same distance, 24 kilometers.
When we calculate the time, it's essential to consider the impact of the current.

• Upstream, the boat works against the current, reducing its effective speed to \(x-2\) km/h.
• Downstream, the boat benefits from the current, increasing its speed to \(x+2\) km/h.

The fundamental formula used here is \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}.\)This formula helps us figure out how long each leg of the journey takes—both against and with the current.

Rational Expressions

Rational expressions are fractions that have polynomials in the numerator, denominator, or both. You see them often in algebra when dealing with formulas and equations involving rates, like our speed and distance problem.
This exercise uses rational expressions to represent the time taken for each part of the trip:

• Upstream: \(\frac{24}{x-2}\)
• Downstream: \(\frac{24}{x+2}\)

These expressions show how the speed of the boat affects the time taken, with different speeds upstream and downstream due to the current.
To solve such problems, you need to be comfortable with operations involving rational expressions, such as addition, as we add the times to get the total round trip time.Understanding how rational expressions relate to real-world problems will help make these concepts clearer while solving problems.

Modeling with Algebra

Modeling with algebra is about creating equations, known as models, that represent real-life situations. In the exercise, we're asked to create an algebraic model for the boat's round trip, considering both directions of travel.
We combine the rational expressions for upstream and downstream travel. This results in the formula:\[\text{Total Time} = \frac{24}{x-2} + \frac{24}{x+2}\]This algebraic model simplifies the complex situation into something easier to analyze and solve.
Such models are invaluable for predicting outcomes, optimizing performance, or understanding patterns. In this case, it shows us how changes in speed \(x\), like increasing or decreasing velocity due to the current, impact the overall travel time.
By applying algebra, you turn real-world observations into mathematical language that can be manipulated and solved.