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Answer :
The area of sector AOB in the circle with OA = 5 and length of arc AB = 1/4 of the circumference is approximately 19.6 square units. (Option a)
To find the area of sector AOB, we need to follow these steps:
Step 1: Calculate the circumference of the circle.
The formula for the circumference of a circle is C = 2πr, where r is the radius. Given that the length of arc AB is 1/4th of the circumference, we can set up the equation:
1/4 * C = 14
Step 2: Find the circumference.
We can rearrange the equation to find C:
C = 14 * 4 = 56
Step 3: Find the radius of the circle.
The circumference of a circle is also given by C = 2πr. We know C = 56, so we can solve for the radius:
56 = 2πr
r = 56 / (2π) ≈ 8.91
Step 4: Calculate the area of the sector.
The area of a sector is given by the formula A = (θ/360) * π * r^2, where θ is the central angle in degrees. In this case, the central angle is 90 degrees since arc AB is 1/4th of the circumference (360 degrees).
A = (90/360) * π * (8.91)^2
Step 5: Calculate the final answer.
Now, substitute the value of π (approximately 3.14) and compute the area:
A ≈ (1/4) * 3.14 * (8.91)^2
A ≈ 0.25 * 3.14 * 79.28
A ≈ 19.82
Step 6: Choose the closest answer.
From the given options, the closest answer to the calculated area is 19.6 square units (option a).
So, the final answer is approximately 19.6 square units (option a).
To learn more about area of sector click here: brainly.com/question/27799926
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