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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. [tex]\( 3.14 \)[/tex]
B. [tex]\( 3.35 \)[/tex]
C. [tex]\( 3.62 \)[/tex]
D. [tex]\( 3.77 \)[/tex]
E. [tex]\( 3.85 \)[/tex]



Answer :

GinnyAnswer
To determine the measure of the central angle in radians for a sector where the ratio of the area of the sector to the area of the circle is [tex]\(\frac{3}{5}\)[/tex], follow these steps:

1. Understanding the Relationship: The ratio of the area of a sector to the total area of the circle is equal to the ratio of the central angle of the sector to the full angle of the circle (which is [tex]\(2\pi\)[/tex] radians).

[tex]\[ \text{Ratio of sector area to total area} = \frac{\text{Central angle}}{2\pi} \][/tex]

2. Given Ratio: We are given that the ratio of the area of sector [tex]\(AOB\)[/tex] to the area of the circle is [tex]\(\frac{3}{5}\)[/tex].

[tex]\[ \frac{\theta}{2\pi} = \frac{3}{5} \][/tex]

3. Solving for [tex]\(\theta\)[/tex]:

Multiply both sides of the equation by [tex]\(2\pi\)[/tex] to isolate [tex]\(\theta\)[/tex]:

[tex]\[ \theta = 2\pi \times \frac{3}{5} \][/tex]

4. Calculate [tex]\(\theta\)[/tex]:

Substituting the value of [tex]\(\pi \approx 3.14159\)[/tex],

[tex]\[ \theta = 2 \times 3.14159 \times \frac{3}{5} = 6.28318 \times \frac{3}{5} = 6.28318 \times 0.6 = 3.76991 \][/tex]

5. Rounding:

Can round the result to two decimal places:

[tex]\[ \theta \approx 3.77 \][/tex]

6. Comparing with Options:

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

7. Conclusion:

Therefore, the correct choice is [tex]\( \boxed{3.77} \)[/tex].

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