A Gardener Uses 144 Feet Of Fencing To Build A Rectangular Garden. The Function $A(w) = 24w - W^2$ Can Be Used To Find The Area Of The Garden For A Given Width, $w$.Does The Function Model The Situation Over The Interval $12 \

Introduction

As a gardener, one of the most crucial decisions is determining the optimal size of the garden. With a limited amount of fencing, the gardener must carefully consider the dimensions of the garden to maximize the available space. In this scenario, we are given 144 feet of fencing to build a rectangular garden. The function $A(w) = 24w - w^2$ is provided to find the area of the garden for a given width, $w$. However, before we can rely on this function to model the situation, we need to determine if it accurately represents the area of the garden over the interval $12 \leq w \leq 24$.

Understanding the Function

The given function $A(w) = 24w - w^2$ represents the area of the garden in terms of the width, $w$. To understand the behavior of this function, let's analyze its components. The first term, $24w$, represents the area of the garden if it were a rectangle with a fixed length of 24 feet and a variable width, $w$. The second term, $-w^2$, represents the area of the garden if it were a rectangle with a fixed width of $w$ and a variable length, $w$. The negative sign indicates that as the width increases, the length decreases, and vice versa.

Graphing the Function

To visualize the behavior of the function, let's graph it over the interval $12 \leq w \leq 24$. We can use a graphing calculator or software to plot the function.

import numpy as np
import matplotlib.pyplot as plt

# Define the function
def A(w):
    return 24*w - w**2

# Generate x values
w = np.linspace(12, 24, 100)

# Calculate corresponding y values
y = A(w)

# Plot the function
plt.plot(w, y)
plt.xlabel('Width (w)')
plt.ylabel('Area (A)')
plt.title('Area of the Garden vs. Width')
plt.grid(True)
plt.show()

Analyzing the Graph

From the graph, we can observe that the function is a downward-facing parabola. This indicates that as the width increases, the area of the garden decreases. The vertex of the parabola is located at $w = 12$, which corresponds to the maximum area of the garden.

Determining the Validity of the Model

To determine if the function accurately models the situation, we need to check if it satisfies the given conditions. The function is defined for all values of $w$, but we need to ensure that it is valid over the interval $12 \leq w \leq 24$. Let's analyze the function at the endpoints of the interval.

Checking the Endpoints

At $w = 12$, the function evaluates to $A(12) = 24(12) - 12^2 = 288 - 144 = 144$. This corresponds to the area of the garden when the width is 12 feet.

At $w = 24$, the function evaluates to $A(24) = 24(24) - 24^2 = 576 - 576 = 0$. This corresponds to the area of the garden when the width is 24 feet.

Conclusion

Based on the analysis, we can conclude that the function $A(w) = 24w - w^2$ accurately models the area of the garden over the interval $12 \leq w \leq 24$. The function is a downward-facing parabola, indicating that as the width increases, the area of the garden decreases. The vertex of the parabola is located at $w = 12$, which corresponds to the maximum area of the garden. Therefore, the gardener can rely on this function to determine the optimal width of the garden to maximize the available space.

Recommendations

Based on the analysis, we recommend the following:

• The gardener should aim to maintain a width of 12 feet to maximize the area of the garden.
• If the gardener wants to increase the length of the garden, they should decrease the width accordingly to maintain the same area.
• The gardener should use the function $A(w) = 24w - w^2$ to determine the area of the garden for different widths.

Future Work

In the future, we can explore other scenarios where the gardener needs to determine the optimal size of the garden. We can also investigate other functions that model the area of the garden and compare their performance with the given function.

References

• [1] "Mathematics for Gardeners". By Amy Stewart. 2014.
• [2] "Gardening Mathematics". By David A. Smith. 2017.

Note: The references provided are fictional and used for demonstration purposes only.

Introduction

In our previous article, we explored the function $A(w) = 24w - w^2$ to model the area of a rectangular garden. We analyzed the function, graphed it, and determined its validity over the interval $12 \leq w \leq 24$. In this article, we will address some common questions and concerns that gardeners may have when using this function to determine the optimal size of their garden.

Q: What is the maximum area of the garden?

A: The maximum area of the garden is achieved when the width is 12 feet. At this width, the area of the garden is 144 square feet.

Q: How does the function account for the length of the garden?

A: The function $A(w) = 24w - w^2$ accounts for the length of the garden by subtracting the square of the width from the product of the width and a fixed length of 24 feet. This represents the area of the garden if it were a rectangle with a fixed length of 24 feet and a variable width, $w$.

Q: Can I use this function to determine the optimal size of my garden if I have a different amount of fencing?

A: No, this function is specifically designed for a garden with 144 feet of fencing. If you have a different amount of fencing, you will need to use a different function to determine the optimal size of your garden.

Q: How do I use this function to determine the area of my garden?

A: To use this function, simply plug in the width of your garden into the equation $A(w) = 24w - w^2$. This will give you the area of your garden in square feet.

Q: What if I want to increase the length of my garden?

A: If you want to increase the length of your garden, you will need to decrease the width accordingly to maintain the same area. You can use the function $A(w) = 24w - w^2$ to determine the new width and length of your garden.

Q: Can I use this function to determine the optimal size of my garden if I have a non-rectangular shape?

A: No, this function is specifically designed for a rectangular garden. If you have a non-rectangular shape, you will need to use a different function or method to determine the optimal size of your garden.

Q: How accurate is this function?

A: The function $A(w) = 24w - w^2$ is an accurate model of the area of a rectangular garden over the interval $12 \leq w \leq 24$. However, it is not a perfect model, and there may be small errors or variations in the actual area of the garden.

Q: Can I use this function to determine the optimal size of my garden if I have a different type of garden, such as a circular or triangular garden?

A: No, this function is specifically designed for a rectangular garden. If you have a different type of garden, you will need to use a different function or method to determine the optimal size of your garden.

Conclusion

In this article, we addressed some common questions and concerns that gardeners may have when using the function $A(w) = 24w - w^2$ to determine the optimal size of their garden. We hope that this article has provided you with a better understanding of how to use this function and how to determine the optimal size of your garden.

Recommendations

• Use the function $A(w) = 24w - w^2$ to determine the optimal size of your rectangular garden.
• Make sure to use the correct amount of fencing for your garden.
• Consider using a different function or method if you have a non-rectangular shape or a different type of garden.

Future Work

In the future, we can explore other scenarios where gardeners need to determine the optimal size of their garden. We can also investigate other functions that model the area of the garden and compare their performance with the given function.

References

• [1] "Mathematics for Gardeners". By Amy Stewart. 2014.
• [2] "Gardening Mathematics". By David A. Smith. 2017.

Note: The references provided are fictional and used for demonstration purposes only.