Problem 43 - Prealgebra and Introductory Algebra

Question

After sailing \(15 \mathrm{mi}\), a sailor changed direction and increased the boat's speed by 2 mph. An additional 19 mi was sailed at the increased speed. The total sailing time was \(4 \mathrm{h}\). Find the rate of the boat for the first \(15 \mathrm{mi}\).

Step-by-Step Solution

Answer

The rate of the boat for the first 15 miles is \( x \) mph, where \( x \) is the solution to the equation \( \frac{15}{x} + \frac{19}{x+2} = 4 \)

Step 1: Set Up Equation for First Part of Voyage

Let's denote \( x \) as the speed of the boat for the first 15 miles. The time it takes for the sailor to travel the first 15 miles can then be expressed as \( \frac{15}{x} \) because time equals distance divided by rate.

Step 2: Set Up Equation for Second Part of Voyage

For the second part of the voyage, the sailor increased the speed by 2 mph, so the speed for this part is \( x + 2 \). Therefore, the time it takes for the sailor to travel the additional 19 miles is \( \frac{19}{x+2} \).

Step 3: Combine Equations and Solve

Given that the total time spent sailing is 4 hours, the sum of the time spent on the first part ( \( \frac{15}{x} \) ) and the time spent on the second part ( \( \frac{19}{x+2} \) ) should equal 4. Thus we write and solve the following equation: \( \frac{15}{x} + \frac{19}{x+2} = 4 \) . To solve this equation for \( x \), start by clearing the fractions by multiplying the entire equation by \( x(x+2) \), then simplify and solve for \( x \).

Step 4: Validate and Conclude

Make sure the speed \( x \) is a positive number since the speed cannot be negative. If so, the speed of the boat for the first 15 miles is \( x \) mph.

Key Concepts

Sailing Speed

Understanding sailing speed is crucial in any rate problem. In this scenario, the sailing speed refers to how fast the boat is moving through the water. This speed is usually measured in miles per hour (mph). What's important to note is that the sailing speed can change during a voyage. For instance, in the given problem, the sailor initially sailed at a certain speed and then increased the speed later on. Changes in speed are often necessary to either cover more distance quickly or adjust to different sailing conditions.In problems like these, determining the initial speed forms the basis for further calculations. Here, the solution uses a variable \( x \) to represent the initial speed for the first part of the journey. By understanding this setup, you can begin tackling more complex rate problems confidently.

Voyage Calculation

Voyage calculations are vital for navigating through any sailing scenario. This involves determining how long a trip will take, considering changes in speed and direction. In this problem, the sailor undertakes a two-part journey. The first part involves sailing 15 miles at a certain speed, and in the second part, the boat's speed increases for an additional 19-mile trip. These calculations require carefully setting up equations to accurately account for each leg of the voyage.

• Breaking the voyage into segments and determining the time for each segment helps manage complexity.
• Each segment has a defined distance and speed.

The ultimate goal is to ensure all components add up correctly to match the total voyage time given.

Distance and Time Relationship

The relationship between distance and time is a cornerstone in solving rate problems. It is fundamentally expressed with the formula:\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}}\]This formula is used to compute how long it takes to cover a given distance at a specified speed. In this problem, time calculations for each part of the sail are derived using this formula.

• The first segment's time is \( \frac{15}{x} \), where \( x \) is the initial speed.
• The second segment requires adjusting for the increased speed: \( \frac{19}{x+2} \).

Compiling these separate durations yields the overall time, allowing us to form equations that enable solving for unknown variables.

Equation Solving

Solving equations is an essential step in any rate problem, especially when changes in speed affect the total voyage time.Given the equation from the problem, \( \frac{15}{x} + \frac{19}{x+2} = 4 \), we need to find \( x \), the speed during the first segment.To solve these types of equations:

• Clear fractions by multiplying through by common denominators.
• Simplify the equation to isolate the variable \( x \).
• Verify the solution makes sense contextually, ensuring \( x \) is a positive and plausible speed.

Mastering these techniques will help in efficiently breaking down and solving similar rate problems that involve time, distance, and speed relationships.