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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?

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?

Elementary Geometry For College Students, 7e
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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**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. \(31.7 \, \text{ft}^2\)
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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