In a circle centered at point [tex]O[/tex], the ratio of the area of sector [tex]AOB[/tex] to the area of the circle is [tex]\frac{3}{5}[/tex]. What is the approximate measure, in radians, of the central angle corresponding to [tex]\widehat{AB}[/tex]? Round the answer to two decimal places.

A. 3.14
B. 3.35
C. 3.62
D. 3.77
E. 3.85



Answer :

To solve this problem, follow these steps:

1. Understand the relationship: The problem states that the ratio of the area of sector [tex]\( AOB \)[/tex] to the area of the entire circle is [tex]\( \frac{3}{5} \)[/tex].

2. Formula for the area of a circle: Recall that the area of a circle is given by [tex]\( \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle.

3. Formula for the area of a sector: The area of a sector of a circle with central angle [tex]\( \theta \)[/tex] (in radians) is given by [tex]\( \frac{\theta}{2\pi} \cdot \pi r^2 \)[/tex].

4. Set up the equation: According to the given ratio [tex]\( \frac{3}{5} \)[/tex], we can write:
[tex]\[ \frac{\text{Area of sector } AOB}{\text{Area of circle}} = \frac{3}{5} \][/tex]
Substituting the formulas for the areas, we get:
[tex]\[ \frac{\frac{\theta}{2\pi} \cdot \pi r^2}{\pi r^2} = \frac{3}{5} \][/tex]

5. Simplify the equation: The [tex]\( \pi r^2 \)[/tex] terms cancel out, leaving us with:
[tex]\[ \frac{\theta}{2\pi} = \frac{3}{5} \][/tex]

6. Solve for [tex]\( \theta \)[/tex]: To isolate [tex]\( \theta \)[/tex], multiply both sides by [tex]\( 2\pi \)[/tex]:
[tex]\[ \theta = \frac{3}{5} \cdot 2\pi \][/tex]

7. Calculate the value of [tex]\( \theta \)[/tex]:
[tex]\[ \theta = \frac{3}{5} \cdot 2\pi \approx 3.7699111843077517 \][/tex]

8. Round the answer: Round the value of [tex]\( \theta \)[/tex] to two decimal places:
[tex]\[ \theta \approx 3.77 \][/tex]

The approximate measure of the central angle corresponding to [tex]\( \widehat{AB} \)[/tex], rounded to two decimal places, is [tex]\( 3.77 \)[/tex] radians.

Thus, the correct answer is:
[tex]\[ \boxed{3.77} \][/tex]