The Harbormaster Wants To Place Buoys Where The River Bottom Is 20 Feet Below The Surface Of The Water. Complete The Absolute Value Equation To Find The Horizontal Distance From The Left Shore At Which The Buoys Should Be Placed.$|\frac{1}{5}s -

Feb 28, 2025 by ADMIN 246 views

The Harbormaster's Buoy Placement Problem: A Mathematical Solution

As a harbormaster, ensuring the safety of vessels and preventing collisions is of utmost importance. One way to achieve this is by strategically placing buoys in the water. However, the placement of these buoys requires careful consideration of the river's depth and the distance from the shore. In this article, we will explore how to use absolute value equations to determine the horizontal distance from the left shore at which the buoys should be placed.

The harbormaster wants to place buoys where the river bottom is 20 feet below the surface of the water. The equation representing the depth of the water is given by:

|\frac{1}{5}s - 20| = 20

where $s$ is the horizontal distance from the left shore.

Before we proceed with solving the equation, let's take a moment to understand what absolute value equations represent. The absolute value of a number is its distance from zero on the number line. In other words, it is the magnitude of the number without considering its direction.

In the given equation, the absolute value expression $|\frac{1}{5}s - 20|$ represents the distance between the depth of the water and the desired depth of 20 feet. The equation states that this distance is equal to 20 feet.

To solve the absolute value equation, we need to consider two cases:

Case 1: $\frac{1}{5}s - 20 \geq 0$

In this case, the absolute value expression can be rewritten as:

\frac{1}{5}s - 20 = 20

Solving for $s$ , we get:

\frac{1}{5}s = 40

s = 200

Case 2: $\frac{1}{5}s - 20 < 0$

In this case, the absolute value expression can be rewritten as:

-(\frac{1}{5}s - 20) = 20

Simplifying, we get:

-\frac{1}{5}s + 20 = 20

-\frac{1}{5}s = 0

s = 0

We have found two possible values for $s$ : 200 and 0. However, we need to consider the context of the problem. The harbormaster wants to place buoys where the river bottom is 20 feet below the surface of the water. This means that the buoys should be placed at a distance from the left shore where the depth of the water is 20 feet.

Based on the absolute value equation, we have found that the buoys should be placed at a horizontal distance of 200 feet from the left shore. This is the solution to the problem posed by the harbormaster.

The use of absolute value equations in real-world problems is vast and varied. In addition to the harbormaster's buoy placement problem, absolute value equations can be used to model a wide range of phenomena, including:

Distance and speed: Absolute value equations can be used to model the distance traveled by an object given its speed and time.
Financial transactions: Absolute value equations can be used to model the cost of a financial transaction given the amount of money involved and the interest rate.
Physical systems: Absolute value equations can be used to model the behavior of physical systems, such as the motion of a pendulum or the flow of a fluid.

In conclusion, the use of absolute value equations is a powerful tool for solving real-world problems. By understanding the concept of absolute value and how to solve absolute value equations, we can model a wide range of phenomena and make informed decisions. Whether it's the harbormaster's buoy placement problem or a more complex real-world problem, absolute value equations provide a powerful framework for solving equations and making predictions.
The Harbormaster's Buoy Placement Problem: A Mathematical Solution - Q&A

In our previous article, we explored how to use absolute value equations to determine the horizontal distance from the left shore at which the buoys should be placed. In this article, we will answer some frequently asked questions related to the problem and provide additional insights.

Q: What is the significance of the absolute value equation in this problem?

A: The absolute value equation represents the distance between the depth of the water and the desired depth of 20 feet. By solving the equation, we can determine the horizontal distance from the left shore at which the buoys should be placed.

Q: Why is it necessary to consider two cases when solving the absolute value equation?

A: When solving absolute value equations, we need to consider two cases: when the expression inside the absolute value is non-negative and when it is negative. This is because the absolute value function behaves differently in these two cases.

Q: What is the difference between the two solutions obtained in the two cases?

A: The two solutions obtained in the two cases represent different scenarios. In the first case, the depth of the water is greater than or equal to 20 feet, while in the second case, the depth of the water is less than 20 feet. The solution obtained in the first case is the one that is relevant to the problem.

Q: Can you explain why the solution obtained in the first case is the one that is relevant to the problem?

A: The solution obtained in the first case is the one that is relevant to the problem because it represents the scenario where the depth of the water is greater than or equal to 20 feet. This is the scenario that the harbormaster is interested in, as he wants to place buoys where the river bottom is 20 feet below the surface of the water.

Q: How does the solution obtained in this problem relate to real-world applications?

A: The solution obtained in this problem has implications for real-world applications, such as:

Distance and speed: The solution can be used to model the distance traveled by an object given its speed and time.
Financial transactions: The solution can be used to model the cost of a financial transaction given the amount of money involved and the interest rate.
Physical systems: The solution can be used to model the behavior of physical systems, such as the motion of a pendulum or the flow of a fluid.

Q: What are some common mistakes to avoid when solving absolute value equations?

A: Some common mistakes to avoid when solving absolute value equations include:

Failing to consider both cases: When solving absolute value equations, it is essential to consider both cases: when the expression inside the absolute value is non-negative and when it is negative.
Not simplifying the equation: Before solving the equation, it is essential to simplify it by combining like terms and eliminating any unnecessary variables.
Not checking the solution: After obtaining a solution, it is essential to check it to ensure that it is valid and makes sense in the context of the problem.