Question
A homeowner wishes to put a fence around a space in her yard to create a dog pen. The space that she wants to fence in is shown below, where each unit on the graph represents one foot.
How many feet of fencing will the homeowner need to build the dog pen?
graph shows 4 points (2,5) (2,-3) (-8,-3) (-3,2)
(1 point)
Responses
23+109−−−√
feet
23 plus square root of 109 feet
33
feet
33 feet
38
feet
38 feet
23+26−−√
feet
How many feet of fencing will the homeowner need to build the dog pen?
graph shows 4 points (2,5) (2,-3) (-8,-3) (-3,2)
(1 point)
Responses
23+109−−−√
feet
23 plus square root of 109 feet
33
feet
33 feet
38
feet
38 feet
23+26−−√
feet
Answers
GPT-4o mini
To determine how many feet of fencing the homeowner will need to build the dog pen, we first need to calculate the length of the perimeter of the polygon formed by the four points given: (2, 5), (2, -3), (-8, -3), and (-3, 2).
1. **Calculate the lengths of each side of the polygon:**
- **Side from (2, 5) to (2, -3)**:
\[
\text{Length} = |5 - (-3)| = |5 + 3| = 8 \text{ feet}
\]
- **Side from (2, -3) to (-8, -3)**:
\[
\text{Length} = |2 - (-8)| = |2 + 8| = 10 \text{ feet}
\]
- **Side from (-8, -3) to (-3, 2)**:
\[
\text{Length} = \sqrt{((-3) - (-8))^2 + (2 - (-3))^2} = \sqrt{(5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \text{ feet}
\]
- **Side from (-3, 2) to (2, 5)**:
\[
\text{Length} = \sqrt{(2 - (-3))^2 + (5 - 2)^2} = \sqrt{(5)^2 + (3)^2} = \sqrt{25 + 9} = \sqrt{34} \text{ feet}
\]
2. **Add these lengths together** to find the total perimeter:
\[
\text{Perimeter} = 8 + 10 + 5\sqrt{2} + \sqrt{34}
\]
Since the closest answer choices provided do not include this exact calculation, let's provide the numerical approximation for possible answer selection:
- The approximate values are as follows:
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{34} \approx 5.831 \)
Calculating the total:
-
\[
\text{Perimeter} \approx 8 + 10 + (5 \times 1.414) + 5.831 \approx 8 + 10 + 7.07 + 5.831 \approx 30.901 \text{ feet}
\]
This rounds up to approximately 31 feet, which isn't listed.
Thus, the best matching estimate from the provided options is:
**33 feet**.
33 feet would be the best choice from the provided responses.
1. **Calculate the lengths of each side of the polygon:**
- **Side from (2, 5) to (2, -3)**:
\[
\text{Length} = |5 - (-3)| = |5 + 3| = 8 \text{ feet}
\]
- **Side from (2, -3) to (-8, -3)**:
\[
\text{Length} = |2 - (-8)| = |2 + 8| = 10 \text{ feet}
\]
- **Side from (-8, -3) to (-3, 2)**:
\[
\text{Length} = \sqrt{((-3) - (-8))^2 + (2 - (-3))^2} = \sqrt{(5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \text{ feet}
\]
- **Side from (-3, 2) to (2, 5)**:
\[
\text{Length} = \sqrt{(2 - (-3))^2 + (5 - 2)^2} = \sqrt{(5)^2 + (3)^2} = \sqrt{25 + 9} = \sqrt{34} \text{ feet}
\]
2. **Add these lengths together** to find the total perimeter:
\[
\text{Perimeter} = 8 + 10 + 5\sqrt{2} + \sqrt{34}
\]
Since the closest answer choices provided do not include this exact calculation, let's provide the numerical approximation for possible answer selection:
- The approximate values are as follows:
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt{34} \approx 5.831 \)
Calculating the total:
-
\[
\text{Perimeter} \approx 8 + 10 + (5 \times 1.414) + 5.831 \approx 8 + 10 + 7.07 + 5.831 \approx 30.901 \text{ feet}
\]
This rounds up to approximately 31 feet, which isn't listed.
Thus, the best matching estimate from the provided options is:
**33 feet**.
33 feet would be the best choice from the provided responses.