Sand Dune Erosion: Solving The Height Equation

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Hey guys! Let's dive into a cool math problem that mixes real-world scenarios with some equation solving. We're talking about a sand dune, that awesome pile of sand you see at the beach, and how it's slowly changing over time. Specifically, this sand dune stands 5 feet above sea level. However, the dune's height isn't static. It's constantly being shaped by the wind, rain, and other natural forces, leading to erosion. The problem tells us the dune is eroding at a rate of 1/20 foot per year. Our goal? To figure out which equation best represents the dune's height after a certain number of years. This is a classic example of a linear equation, where we're looking at a constant rate of change. So, let's break it down and see how we can solve this together.

Understanding the Problem: The Sand Dune's Tale

Okay, so the sand dune starts at a height of 5 feet. That's our starting point, our initial value. Think of it like this: on day zero, before any erosion happens, the dune is at its full height. Then, every year, a little bit of the dune disappears due to erosion. The erosion rate is 1/20 foot per year. This means that every year, the dune gets a tiny bit shorter. The rate of erosion is a constant value, meaning it doesn't change from year to year. This constant rate of erosion is key to understanding the equation. Now, we want to write an equation that describes the height of the dune (y) after a certain number of years (x). We're essentially tracking how the dune's height changes over time.

To make this super clear, imagine this: after 1 year, the dune will be 1/20 foot shorter. After 2 years, it will be 2/20 foot shorter, and so on. The erosion is a consistent process, and we can model it mathematically. We are going to represent the decrease in height with a negative sign. Because the dune is losing height, we will subtract the amount of erosion from the initial height. That negative rate of change is absolutely essential in the equation.

Breaking Down the Equation: Unraveling the Variables

Alright, let's look at the options provided. To figure out the right equation, we need to understand the components of a linear equation. The general form of a linear equation is y = mx + b, where:

  • y represents the dependent variable (in this case, the height of the dune).
  • x represents the independent variable (the number of years).
  • m represents the slope, which is the rate of change (erosion rate in our case).
  • b represents the y-intercept, which is the initial value (the starting height of the dune).

In our problem:

  • The initial height (b) of the sand dune is 5 feet. This is where the dune starts, its height before any erosion.
  • The erosion rate (m) is -1/20 foot per year. It's negative because the dune's height is decreasing. So, with each passing year, the dune gets a little shorter. The rate of change is the key to creating the proper equation.

Now, let's apply this information to the general form of the linear equation (y = mx + b). We know 'm' is -1/20, and 'b' is 5. So, the equation should be y = (-1/20)x + 5. By knowing what each part of the equation represents, we can easily find the right answer. We're looking for an equation that accurately reflects the dune's changing height, so we must make sure all the signs and numbers are correct.

Examining the Answer Choices: Finding the Right Match

Let's evaluate the answer choices one by one:

  • A. y = -1/20x - 5: This equation uses the correct slope (-1/20, representing the erosion rate). However, it incorrectly subtracts 5, instead of adding it. It suggests that the dune started below sea level, which isn't possible in our scenario. This is not the correct equation because it fails to represent the initial height of the sand dune correctly.
  • B. y = -1/20x + 5: This equation is perfect! It correctly represents the erosion rate (-1/20) and the initial height (+5). This is the best match! The equation correctly shows the dune's height decreasing over time, starting from 5 feet and eroding at a rate of 1/20 foot per year. This correctly models the problem scenario.
  • C. y = -20x - 5: This equation has the correct negative sign but the incorrect slope (-20). This would imply an extremely rapid erosion rate, which isn't true for our problem. It also incorrectly subtracts the initial height.
  • D. y = -20x + 5: This has the incorrect slope (-20) but includes the correct initial height (+5). The rate of erosion is off, which doesn't reflect the situation. This suggests the dune is losing height at a much faster rate than 1/20 foot per year. Incorrect! This isn't the correct representation of the problem's scenario.

So, after evaluating the choices, we can confidently say that option B. y = -1/20x + 5 is the correct equation. It accurately portrays the dune's height, taking into account the initial height and the erosion rate.

The Final Answer: The Equation's Victory

Therefore, the equation that represents the height of the sand dune after x years is B. y = -1/20x + 5. This equation allows us to calculate the height of the dune at any given point in time. It's a simple, elegant way to model a real-world phenomenon. Isn't that cool?

So, to recap: we've broken down the problem step by step, identified the key variables, and used them to create the correct equation. We've understood how erosion affects the height of the sand dune over time, and that the initial height, erosion rate, and the passage of time all work together to model this scenario. Now, you should be able to solve similar problems. Keep practicing, and you will become experts at solving these types of equations!