A Sand Dune Stands 5 Feet Above Sea Level. The Hill Is Eroding At A Rate Of 1 Foot Per 20 Years. Let Y Y Y Represent The Height Of The Sand Dune After X X X Years. Which Equation Represents The Situation?A. Y = − 1 20 X − 5 Y=-\frac{1}{20} X-5 Y = − 20 1 X − 5
Introduction
The natural world is full of fascinating phenomena that can be modeled using mathematical equations. One such phenomenon is the erosion of a sand dune over time. In this article, we will explore how to represent the situation mathematically, using the given information about the sand dune's height and erosion rate.
Understanding the Problem
We are given that a sand dune stands 5 feet above sea level and is eroding at a rate of 1 foot per 20 years. We need to find an equation that represents the height of the sand dune after x years. Let's denote the height of the sand dune as y and the time in years as x.
Identifying the Key Components
To model the situation mathematically, we need to identify the key components:
- The initial height of the sand dune: 5 feet
- The erosion rate: 1 foot per 20 years
- The time variable: x years
Developing the Equation
We know that the sand dune is eroding at a rate of 1 foot per 20 years. This means that for every 20 years, the height of the sand dune decreases by 1 foot. To represent this mathematically, we can use the concept of a linear equation.
A linear equation has the form y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope (m) represents the erosion rate, and the y-intercept (b) represents the initial height of the sand dune.
Calculating the Slope
The slope (m) represents the change in height (y) per unit change in time (x). In this case, the erosion rate is 1 foot per 20 years, so the slope is:
m = -1/20
The negative sign indicates that the sand dune is eroding, i.e., its height is decreasing over time.
Calculating the Y-Intercept
The y-intercept (b) represents the initial height of the sand dune, which is 5 feet.
b = 5
Writing the Equation
Now that we have the slope (m) and the y-intercept (b), we can write the equation:
y = -1/20 x + 5
This equation represents the height of the sand dune after x years.
Conclusion
In this article, we have developed a mathematical equation to represent the erosion of a sand dune over time. We have identified the key components, including the initial height of the sand dune, the erosion rate, and the time variable. We have then used the concept of a linear equation to model the situation mathematically. The resulting equation is:
y = -1/20 x + 5
This equation can be used to predict the height of the sand dune after any given number of years.
Discussion
The equation y = -1/20 x + 5 represents a linear relationship between the height of the sand dune (y) and the time (x). The slope (m) of -1/20 indicates that the sand dune is eroding at a rate of 1 foot per 20 years. The y-intercept (b) of 5 represents the initial height of the sand dune.
Comparison with the Given Options
The equation y = -1/20 x + 5 is one of the options provided. Let's compare it with the other options:
A. y = -1/20 x - 5
This option is incorrect because the y-intercept is -5, which is not the initial height of the sand dune.
B. y = -1/20 x + 10
This option is incorrect because the y-intercept is 10, which is not the initial height of the sand dune.
C. y = -1/20 x + 5
This option is correct because it matches the equation we have developed.
Final Answer
The final answer is:
y = -1/20 x + 5
Introduction
In our previous article, we explored how to represent the erosion of a sand dune over time using a mathematical equation. We developed the equation y = -1/20 x + 5, where y represents the height of the sand dune and x represents the time in years.
In this article, we will answer some frequently asked questions about the sand dune's erosion and the mathematical equation that represents it.
Q&A
Q: What is the initial height of the sand dune?
A: The initial height of the sand dune is 5 feet.
Q: What is the erosion rate of the sand dune?
A: The erosion rate of the sand dune is 1 foot per 20 years.
Q: What does the slope (m) represent in the equation y = -1/20 x + 5?
A: The slope (m) represents the erosion rate of the sand dune, which is 1 foot per 20 years.
Q: What does the y-intercept (b) represent in the equation y = -1/20 x + 5?
A: The y-intercept (b) represents the initial height of the sand dune, which is 5 feet.
Q: How can we use the equation y = -1/20 x + 5 to predict the height of the sand dune after a given number of years?
A: To predict the height of the sand dune after a given number of years, we can plug in the value of x into the equation y = -1/20 x + 5. For example, if we want to know the height of the sand dune after 10 years, we can plug in x = 10 into the equation.
Q: What happens to the height of the sand dune over time?
A: The height of the sand dune decreases over time due to erosion.
Q: Is the equation y = -1/20 x + 5 a linear equation?
A: Yes, the equation y = -1/20 x + 5 is a linear equation because it has the form y = mx + b, where m is the slope and b is the y-intercept.
Q: Can we use the equation y = -1/20 x + 5 to model other types of erosion?
A: Yes, we can use the equation y = -1/20 x + 5 as a model for other types of erosion, such as the erosion of a cliff or a mountain.
Conclusion
In this article, we have answered some frequently asked questions about the sand dune's erosion and the mathematical equation that represents it. We have also discussed how to use the equation to predict the height of the sand dune after a given number of years.
Discussion
The equation y = -1/20 x + 5 is a simple yet powerful tool for modeling the erosion of a sand dune. By understanding the slope and y-intercept of the equation, we can gain insights into the behavior of the sand dune over time.
Final Answer
The final answer is:
y = -1/20 x + 5