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Find the area of a sector with a central angle of 160° and a dlameter of 8.6 cm. Round to the nearest tenth. O 7.6 cm2 O 25.8 cm2 O 103.3 cm2 O 3 cm?

Find the area of a sector with a central angle of 160° and a dlameter of 8.6 cm. Round to the nearest tenth. O 7.6 cm2 O 25.8 cm2 O 103.3 cm2 O 3 cm?

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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**Question:** Find the area of a sector with a central angle of 160° and a diameter of 8.6 cm. Round to the nearest tenth. **Options:** - ○ 7.6 cm² - ○ 25.8 cm² - ○ 103.3 cm² - ○ 3 cm² **Explanation:** To find the area of a sector, you can use the formula: \[ \text{Area of sector} = \left( \frac{\theta}{360} \right) \times \pi \times r^2 \] Where: - \(\theta\) is the central angle in degrees - \(r\) is the radius of the circle Given: - Central angle (\(\theta\)) = 160° - Diameter = 8.6 cm First, calculate the radius \(r\): \[ r = \frac{\text{diameter}}{2} = \frac{8.6}{2} = 4.3 \text{ cm} \] Then plug these values into the formula: \[ \text{Area of sector} = \left( \frac{160}{360} \right) \times \pi \times (4.3)^2 \] Simplify the fraction: \[ \left( \frac{160}{360} \right) = \left( \frac{4}{9} \right) \] Now calculate: \[ \text{Area of sector} = \left(\frac{4}{9}\right) \times \pi \times 18.49 \] \[ \text{Area of sector} \approx \left(\frac{4}{9}\right) \times 3.1416 \times 18.49 \] \[ \text{Area of sector} \approx \left(\frac{4}{9}\right) \times 58.09 \] \[ \text{Area of sector} \approx 25.827 \text{ cm}^2 \] Rounded to the nearest tenth: \[ \text{Area of sector} \approx 25.8 \text{ cm}^2 \] Therefore, the correct option is: - ○ 25.8 cm²
Transcribed Image Text:**Question:** Find the area of a sector with a central angle of 160° and a diameter of 8.6 cm. Round to the nearest tenth. **Options:** - ○ 7.6 cm² - ○ 25.8 cm² - ○ 103.3 cm² - ○ 3 cm² **Explanation:** To find the area of a sector, you can use the formula: \[ \text{Area of sector} = \left( \frac{\theta}{360} \right) \times \pi \times r^2 \] Where: - \(\theta\) is the central angle in degrees - \(r\) is the radius of the circle Given: - Central angle (\(\theta\)) = 160° - Diameter = 8.6 cm First, calculate the radius \(r\): \[ r = \frac{\text{diameter}}{2} = \frac{8.6}{2} = 4.3 \text{ cm} \] Then plug these values into the formula: \[ \text{Area of sector} = \left( \frac{160}{360} \right) \times \pi \times (4.3)^2 \] Simplify the fraction: \[ \left( \frac{160}{360} \right) = \left( \frac{4}{9} \right) \] Now calculate: \[ \text{Area of sector} = \left(\frac{4}{9}\right) \times \pi \times 18.49 \] \[ \text{Area of sector} \approx \left(\frac{4}{9}\right) \times 3.1416 \times 18.49 \] \[ \text{Area of sector} \approx \left(\frac{4}{9}\right) \times 58.09 \] \[ \text{Area of sector} \approx 25.827 \text{ cm}^2 \] Rounded to the nearest tenth: \[ \text{Area of sector} \approx 25.8 \text{ cm}^2 \] Therefore, the correct option is: - ○ 25.8 cm²
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