Answer:
Approximately 128.57°
Explanation:
To find the central angle, we can use the formula:
Area of sector = (central angle/360°) * π * r²
Given that the area of the sector is 28 m², we can plug in the known values into the formula:
28 = (central angle/360°) * π * r²
We're given the intercepted arc of the circle, which is 10 m. The length of the intercepted arc is equal to the circumference of the entire circle multiplied by the fraction of the circle that the sector represents. The circumference of a circle is given by the formula:
Circumference = 2πr
We can set up the following equation:
10 = (central angle/360°) * 2π * r
Simplifying, we have:
10 = (central angle/360°) * 2π * r
r = 10 / ((central angle/360°) * 2π)
Substituting this value for r in the area of sector formula, we have:
28 = (central angle/360°) * π * (10 / ((central angle/360°) * 2π))²
Simplifying further, we have:
28 = (central angle/360°) * 100π / (central angle/360°)²
Cross-multiplying, we get:
28 * (central angle/360°)² = (central angle/360°) * 100π
Simplifying and rearranging the equation, we have:
28 * (central angle/360°) = 100π
Dividing both sides by 28, we get:
(central angle/360°) = 100π / 28
Multiplying both sides by 360°, we get:
central angle = 100π / 28 * 360°
Simplifying, we have:
central angle = 128.57°
Therefore, the central angle is approximately 128.57°.