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
![**Geometry Problem: Finding the Area of a Pool Table**
**Problem Statement:**
A pool table has a diagonal that measures 8 feet long and a side that measures 6 feet long. What is the area of the pool table?
**Options:**
a. \(10 \, \text{ft}^2\)
b. \(100 \, \text{ft}^2\)
c. \(5.3 \, \text{ft}^2\)
d. \(31.7 \, \text{ft}^2\)
To solve this problem, we need to use the Pythagorean theorem. Given that the diagonal (hypotenuse) of the right triangle formed by the sides of the pool table is 8 feet and one side (a) is 6 feet, we can find the length of the other side (b):
1. Applying the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Where:
- \(c = 8 \, \text{ft} \) (diagonal)
- \(a = 6 \, \text{ft} \) (one side)
2. Substituting the known values:
\[
6^2 + b^2 = 8^2
\]
\[
36 + b^2 = 64
\]
3. Solving for \(b\):
\[
b^2 = 64 - 36
\]
\[
b^2 = 28
\]
\[
b = \sqrt{28} \approx 5.29 \, \text{ft}
\]
4. Now, we find the area of the rectangle (pool table):
\[
\text{Area} = a \times b = 6 \, \text{ft} \times 5.29 \, \text{ft} \approx 31.74 \, \text{ft}^2
\]
Therefore, the correct answer is:
d. \(31.7 \, \text{ft}^2\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febcd91cf-c92a-4a76-9c9a-e8d21c0f504d%2F7015d0b5-aec8-46cf-a00d-4eddb8c68ced%2Fjcdcxvj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Geometry Problem: Finding the Area of a Pool Table**
**Problem Statement:**
A pool table has a diagonal that measures 8 feet long and a side that measures 6 feet long. What is the area of the pool table?
**Options:**
a. \(10 \, \text{ft}^2\)
b. \(100 \, \text{ft}^2\)
c. \(5.3 \, \text{ft}^2\)
d. \(31.7 \, \text{ft}^2\)
To solve this problem, we need to use the Pythagorean theorem. Given that the diagonal (hypotenuse) of the right triangle formed by the sides of the pool table is 8 feet and one side (a) is 6 feet, we can find the length of the other side (b):
1. Applying the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Where:
- \(c = 8 \, \text{ft} \) (diagonal)
- \(a = 6 \, \text{ft} \) (one side)
2. Substituting the known values:
\[
6^2 + b^2 = 8^2
\]
\[
36 + b^2 = 64
\]
3. Solving for \(b\):
\[
b^2 = 64 - 36
\]
\[
b^2 = 28
\]
\[
b = \sqrt{28} \approx 5.29 \, \text{ft}
\]
4. Now, we find the area of the rectangle (pool table):
\[
\text{Area} = a \times b = 6 \, \text{ft} \times 5.29 \, \text{ft} \approx 31.74 \, \text{ft}^2
\]
Therefore, the correct answer is:
d. \(31.7 \, \text{ft}^2\)
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