Question

in a circle centered at point o, the ratio of the area of sector aob to the area of the circle is \\(\frac{3}{5}\\). what is the approximate measure, in radians, of the central angle corresponding to \\(\overarc{ab}\\)? round the answer to two decimal places.\
a. 3.14\
b. 3.35\
c. 3.62\
d. 3.77\
e. 3.85

Explanation:

Step1: Recall sector area formula

The area of a sector \( A_{sector} = \frac{1}{2}r^2\theta \), and the area of a circle \( A_{circle} = \pi r^2 \). The ratio of the sector area to the circle area is \( \frac{A_{sector}}{A_{circle}}=\frac{\frac{1}{2}r^2\theta}{\pi r^2}=\frac{\theta}{2\pi} \).

Step2: Set up the equation

Given the ratio is \( \frac{3}{5} \), so \( \frac{\theta}{2\pi}=\frac{3}{5} \).

Step3: Solve for \(\theta\)

Multiply both sides by \( 2\pi \): \( \theta=\frac{6\pi}{5} \). Calculate \( \frac{6\pi}{5}\approx\frac{6\times3.1416}{5}\approx3.77 \).

Answer:

D. 3.77