A contractor wants to fence a rectangular yard by using the wall of the house as one side of the rectangle and then enclosing the other three sides with fence. The yard has an area of 600 square feet. Undo Clear All HOUSE Note: Figure not drawn to scale I X YARD 600 ft² x+1 How many feet of fencing are needed? O 24 feet O 25 feet O 73 feet O 98 feet
A contractor wants to fence a rectangular yard by using the wall of the house as one side of the rectangle and then enclosing the other three sides with fence. The yard has an area of 600 square feet. Undo Clear All HOUSE Note: Figure not drawn to scale I X YARD 600 ft² x+1 How many feet of fencing are needed? O 24 feet O 25 feet O 73 feet O 98 feet
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
![### Problem Statement
A contractor wants to fence a rectangular yard by using the wall of the house as one side of the rectangle and then enclosing the other three sides with a fence. The yard has an area of 600 square feet.
### Diagram Explanation
The accompanying diagram shows a house along with the rectangular yard that needs fencing:
- The yard is indicated to have an area of 600 square feet.
- One side of the yard (with length \( x \)) is along the house.
- The other side of the yard has a length of \( x + 1 \).
A note at the bottom of the diagram clarifies that the figure is not drawn to scale.
### Question
How many feet of fencing are needed?
- 24 feet
- 25 feet
- 73 feet
- 98 feet
### Detailed Analysis
To determine the amount of fencing required, we need to calculate the perimeter of the three sides that need fencing (since one side is the wall of the house).
Given:
- The area of the yard: \( 600 \, \text{ft}^2 \)
- Lengths of the sides: \( x \) and \( x + 1 \)
### Calculations
The area of a rectangle is given by:
\[ \text{Area} = \text{length} \times \text{width} \]
So, we have:
\[ x \times (x + 1) = 600 \]
Solving for \( x \):
\[ x^2 + x - 600 = 0 \]
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substitute \( a = 1 \), \( b = 1 \), and \( c = -600 \):
\[ x = \frac{-1 \pm \sqrt{(1)^2 - 4 \cdot 1 \cdot (-600)}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + 2400}}{2} \]
\[ x = \frac{-1 \pm \sqrt{2401}}{2} \]
\[ x = \frac{-1 \pm 49}{2} \]
We get two solutions, but only the positive value is considered:
\[ x = \frac{48}{2} =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf693007-feb9-4e65-ba86-b1b16e800487%2F985962a2-8592-49e2-93b3-9b8a650c42f6%2Fh7pm5nh_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
A contractor wants to fence a rectangular yard by using the wall of the house as one side of the rectangle and then enclosing the other three sides with a fence. The yard has an area of 600 square feet.
### Diagram Explanation
The accompanying diagram shows a house along with the rectangular yard that needs fencing:
- The yard is indicated to have an area of 600 square feet.
- One side of the yard (with length \( x \)) is along the house.
- The other side of the yard has a length of \( x + 1 \).
A note at the bottom of the diagram clarifies that the figure is not drawn to scale.
### Question
How many feet of fencing are needed?
- 24 feet
- 25 feet
- 73 feet
- 98 feet
### Detailed Analysis
To determine the amount of fencing required, we need to calculate the perimeter of the three sides that need fencing (since one side is the wall of the house).
Given:
- The area of the yard: \( 600 \, \text{ft}^2 \)
- Lengths of the sides: \( x \) and \( x + 1 \)
### Calculations
The area of a rectangle is given by:
\[ \text{Area} = \text{length} \times \text{width} \]
So, we have:
\[ x \times (x + 1) = 600 \]
Solving for \( x \):
\[ x^2 + x - 600 = 0 \]
Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substitute \( a = 1 \), \( b = 1 \), and \( c = -600 \):
\[ x = \frac{-1 \pm \sqrt{(1)^2 - 4 \cdot 1 \cdot (-600)}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + 2400}}{2} \]
\[ x = \frac{-1 \pm \sqrt{2401}}{2} \]
\[ x = \frac{-1 \pm 49}{2} \]
We get two solutions, but only the positive value is considered:
\[ x = \frac{48}{2} =
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