**Optimizing the Area of a Rectangular Garden with Limited Fencing**Hans has 360 meters of fencing. He will use it to form three sides of a rectangular garden. The fourth side will be along a house and will not need fencing.**Diagram Description:**The diagram shows a rectangular garden with one side labeled as $ x $ which is aligned along the house. The remaining three sides are fenced, and the length of the two sides perpendicular to $ x $ is unknown.**Problem Statements and Questions:**(a) **Find a function that gives the area $ A(x) $ of the garden (in square meters) in terms of $ x $.**\[ A(x) = \_\_\_\_ \](b) **What side length $ x $ gives the maximum area that the garden can have?**\[ \text{Side length } x = \_\_\_\_ \text{ meters} \](c) **What is the maximum area that the garden can have?**\[ \text{Maximum area} = \_\_\_\_ \text{ square meters} \]**Explanation:**To solve this problem, we need to derive the mathematical function for the area of the rectangular garden using the given fencing condition, optimize that function to find the value of $ x $ that maximizes the area, and finally calculate the maximum area.1. **Deriving the Area Function:** - The total length of the fence used for the three sides is 360 meters. - The garden has one side along the house with length $ x $ and the other two sides are equal in length, let's call this length $ y $. - Therefore, the total fencing used is:\[ x + 2y = 360 \]\[ y = \frac{360 - x}{2} \] - The area $ A $ of the rectangle is given by the product of its length and width:\[ A(x) = x \cdot y \]Substituting the value of $ y $:\[ A(x) = x \cdot \frac{360 - x}{2} \]\[ A(x) = \frac{360x - x^2}{2} \]\[ A(x) = 180x - \frac{x^2}{2} \]2. **Finding the Maximum Area:** - To find the maximum area, we

Total fencing Hans have = 360 m given that the shape of garden is rectangular and only 3 sides are…

Hans has 360 meters of fencing. He will use it to form three sides of a rectangular garden. The fourth side will be along a house and will not need fencing. As shown below, one of the sides has length x (in meters). Side along house (a) Find a function that gives the area A (x) of the garden (in square meters) in terms of x. A(:) = 0 (b) What side length x gives the maximum area that the garden can have? Side length x : meters (c) What is the maximum area that the garden can have? Maximum area: square meters

Hans has 360 meters of fencing. He will use it to form three sides of a rectangular garden. The fourth side will be along a house and will not need fencing. As shown below, one of the sides has length x (in meters). Side along house (a) Find a function that gives the area A (x) of the garden (in square meters) in terms of x. A(:) = 0 (b) What side length x gives the maximum area that the garden can have? Side length x : meters (c) What is the maximum area that the garden can have? Maximum area: square meters

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Question

**Optimizing the Area of a Rectangular Garden with Limited Fencing**

Hans has 360 meters of fencing. He will use it to form three sides of a rectangular garden. The fourth side will be along a house and will not need fencing.

**Diagram Description:**
The diagram shows a rectangular garden with one side labeled as $ x $ which is aligned along the house. The remaining three sides are fenced, and the length of the two sides perpendicular to $ x $ is unknown.

**Problem Statements and Questions:**

(a) **Find a function that gives the area $ A(x) $ of the garden (in square meters) in terms of $ x $.**

\[ A(x) = \_\_\_\_ \]

(b) **What side length $ x $ gives the maximum area that the garden can have?**

\[ \text{Side length } x = \_\_\_\_ \text{ meters} \]

(c) **What is the maximum area that the garden can have?**

\[ \text{Maximum area} = \_\_\_\_ \text{ square meters} \]

**Explanation:**

To solve this problem, we need to derive the mathematical function for the area of the rectangular garden using the given fencing condition, optimize that function to find the value of $ x $ that maximizes the area, and finally calculate the maximum area.

1. **Deriving the Area Function:**
- The total length of the fence used for the three sides is 360 meters.
- The garden has one side along the house with length $ x $ and the other two sides are equal in length, let's call this length $ y $.
- Therefore, the total fencing used is:

\[ x + 2y = 360 \]

\[ y = \frac{360 - x}{2} \]

- The area $ A $ of the rectangle is given by the product of its length and width:

\[ A(x) = x \cdot y \]

Substituting the value of $ y $:

\[ A(x) = x \cdot \frac{360 - x}{2} \]

\[ A(x) = \frac{360x - x^2}{2} \]

\[ A(x) = 180x - \frac{x^2}{2} \]

2. **Finding the Maximum Area:**
- To find the maximum area, we

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