Select The Correct Answer.Sam Is Installing A Walkway Around A Rectangular Flower Patch In His Garden. The Flower Patch Is 12 Feet Long And 6 Feet Wide. The Width Of The Walkway Is $x$ Feet.Sam Created Function $A(x$\] To Represent The

Introduction

When designing a walkway around a rectangular flower patch, it's essential to consider the dimensions of the patch and the width of the walkway. In this scenario, Sam is installing a walkway around a rectangular flower patch that is 12 feet long and 6 feet wide. The width of the walkway is represented by the variable $x$ feet. To determine the total area of the walkway, Sam created a function $A(x)$ to represent the area of the walkway.

Understanding the Function

The function $A(x)$ represents the area of the walkway, which can be calculated by finding the difference between the total area of the walkway and the flower patch. To find the total area of the walkway, we need to add the width of the walkway to the length and width of the flower patch.

Calculating the Total Area of the Walkway

The total area of the walkway can be calculated using the formula:

$$A(x) = (12 + 2x)(6 + 2x)$$

This formula represents the total area of the walkway, which includes the area of the flower patch and the area of the walkway itself.

Simplifying the Function

To simplify the function $A(x)$, we can expand the formula:

$$A(x) = (12 + 2x)(6 + 2x)$$

$$A(x) = 72 + 24x + 12x + 4x^2$$

$$A(x) = 72 + 36x + 4x^2$$

This simplified formula represents the total area of the walkway.

Graphing the Function

To visualize the function $A(x)$, we can graph the function using a graphing calculator or a computer algebra system. The graph of the function will show the relationship between the width of the walkway and the total area of the walkway.

Interpreting the Graph

The graph of the function $A(x)$ will show that the total area of the walkway increases as the width of the walkway increases. The graph will also show that the rate of increase in the total area of the walkway decreases as the width of the walkway increases.

Conclusion

In conclusion, the function $A(x)$ represents the total area of the walkway around a rectangular flower patch. The function can be calculated using the formula $A(x) = (12 + 2x)(6 + 2x)$, which can be simplified to $A(x) = 72 + 36x + 4x^2$. The graph of the function shows the relationship between the width of the walkway and the total area of the walkway.

Key Takeaways

• The function $A(x)$ represents the total area of the walkway around a rectangular flower patch.
• The function can be calculated using the formula $A(x) = (12 + 2x)(6 + 2x)$.
• The graph of the function shows the relationship between the width of the walkway and the total area of the walkway.
• The total area of the walkway increases as the width of the walkway increases.
• The rate of increase in the total area of the walkway decreases as the width of the walkway increases.

Practice Problems

1. Find the total area of the walkway when the width of the walkway is 2 feet.
2. Find the width of the walkway that will result in a total area of 200 square feet.
3. Graph the function $A(x)$ and identify the x-intercept.

Answer Key

1. $A(2) = 72 + 36(2) + 4(2)^2 = 144$
2. $72 + 36x + 4x^2 = 200$ $4x^2 + 36x - 128 = 0$ $x = \frac{-36 \pm \sqrt{36^2 - 4(4)(-128)}}{2(4)}$ $x = \frac{-36 \pm \sqrt{1296 + 2048}}{8}$ $x = \frac{-36 \pm \sqrt{3344}}{8}$ $x = \frac{-36 \pm 57.6}{8}$ $x = \frac{-36 + 57.6}{8}$ or $x = \frac{-36 - 57.6}{8}$ $x = \frac{21.6}{8}$ or $x = \frac{-93.6}{8}$ $x = 2.7$ or $x = -11.7$ Since the width of the walkway cannot be negative, the only solution is $x = 2.7$.
3. The x-intercept of the graph of the function $A(x)$ is the point where the graph intersects the x-axis. To find the x-intercept, we can set the function equal to zero and solve for x:

$$A(x) = 0$$

$$72 + 36x + 4x^2 = 0$$

$$4x^2 + 36x + 72 = 0$$

$$(2x + 18)(2x + 18) = 0$$

$$2x + 18 = 0$$

$$2x = -18$$

$$x = -9$$

$$$$

Q: What is the function A(x) used for?

A: The function A(x) is used to represent the total area of the walkway around a rectangular flower patch. It takes into account the dimensions of the flower patch and the width of the walkway.

Q: How is the function A(x) calculated?

A: The function A(x) is calculated using the formula A(x) = (12 + 2x)(6 + 2x), where x represents the width of the walkway.

Q: What is the simplified form of the function A(x)?

A: The simplified form of the function A(x) is A(x) = 72 + 36x + 4x^2.

Q: What does the graph of the function A(x) show?

A: The graph of the function A(x) shows the relationship between the width of the walkway and the total area of the walkway. It also shows that the total area of the walkway increases as the width of the walkway increases.

Q: What is the x-intercept of the graph of the function A(x)?

A: The x-intercept of the graph of the function A(x) is the point where the graph intersects the x-axis. To find the x-intercept, we can set the function equal to zero and solve for x. The x-intercept is the point (-9, 0).

Q: How do I find the total area of the walkway when the width of the walkway is 2 feet?

A: To find the total area of the walkway when the width of the walkway is 2 feet, we can plug x = 2 into the function A(x) = 72 + 36x + 4x^2. This gives us A(2) = 72 + 36(2) + 4(2)^2 = 144.

Q: How do I find the width of the walkway that will result in a total area of 200 square feet?

A: To find the width of the walkway that will result in a total area of 200 square feet, we can set the function A(x) = 72 + 36x + 4x^2 equal to 200 and solve for x. This gives us 72 + 36x + 4x^2 = 200. We can then solve for x using the quadratic formula.

Q: What is the relationship between the width of the walkway and the total area of the walkway?

A: The relationship between the width of the walkway and the total area of the walkway is shown by the graph of the function A(x). As the width of the walkway increases, the total area of the walkway also increases.

Q: Can the width of the walkway be negative?

A: No, the width of the walkway cannot be negative. The width of the walkway must be a positive value.

Q: What is the significance of the x-intercept of the graph of the function A(x)?

A: The x-intercept of the graph of the function A(x) represents the point where the graph intersects the x-axis. This point is important because it shows the relationship between the width of the walkway and the total area of the walkway.

Q: How can I use the function A(x) in real-world applications?

A: The function A(x) can be used in real-world applications such as designing walkways around gardens, parks, and other outdoor spaces. It can also be used to calculate the total area of a walkway in a variety of different situations.

Q: What are some common mistakes to avoid when using the function A(x)?

A: Some common mistakes to avoid when using the function A(x) include:

• Not considering the dimensions of the flower patch and the width of the walkway
• Not using the correct formula for the function A(x)
• Not solving for x correctly
• Not considering the x-intercept of the graph of the function A(x)

Q: How can I improve my understanding of the function A(x)?

A: To improve your understanding of the function A(x), you can:

• Practice using the function A(x) in different scenarios
• Graph the function A(x) to visualize the relationship between the width of the walkway and the total area of the walkway
• Use real-world examples to apply the function A(x)
• Review the formula for the function A(x) and make sure you understand how it works.