Answered

Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.

Select the correct answer.

Points [tex]$A$[/tex] and [tex]$B$[/tex] lie on a circle centered at point [tex]$O$[/tex]. If [tex]$\frac{\text{length of }\hat{AB}}{\text{radius}} = \frac{\pi}{10}$[/tex], what is the ratio of the area of sector [tex]$AOB$[/tex] to the area of the circle?

A. [tex]$\frac{1}{10}$[/tex]

B. [tex]$\frac{\pi}{10}$[/tex]

C. [tex]$\frac{1}{20}$[/tex]

D. [tex]$\frac{\pi}{20}$[/tex]

Sagot :

GinnyAnswer
To solve the problem, we need to understand the relationship between the arc length and the radius of a circle involving the sector of the circle.

Given:
[tex]\[ \frac{\text{length of }\hat{AB}}{\text{radius}} = \frac{\pi}{10} \][/tex]

We denote:
- [tex]\(l\)[/tex] as the length of arc [tex]\(\hat{AB}\)[/tex]
- [tex]\(r\)[/tex] as the radius of the circle

The formula connecting the arc length [tex]\(l\)[/tex] to the central angle [tex]\(\theta\)[/tex] in radians is:
[tex]\[ l = r \theta \][/tex]

From the given relation, we can equate:
[tex]\[ \frac{l}{r} = \frac{\pi}{10} \][/tex]

So,
[tex]\[ l = r \cdot \frac{\pi}{10} \][/tex]

This implies that the central angle [tex]\(\theta\)[/tex] subtended by the arc [tex]\(\hat{AB}\)[/tex] is:
[tex]\[ \theta = \frac{\pi}{10} \text{ radians} \][/tex]

Next, we calculate the area of sector [tex]\(AOB\)[/tex]. The area [tex]\(A_{\text{sector}}\)[/tex] of a sector with a central angle [tex]\(\theta\)[/tex] (in radians) in a circle of radius [tex]\(r\)[/tex] is given by:
[tex]\[ A_{\text{sector}} = \frac{1}{2} r^2 \theta \][/tex]

Substituting [tex]\(\theta = \frac{\pi}{10}\)[/tex]:
[tex]\[ A_{\text{sector}} = \frac{1}{2} r^2 \cdot \frac{\pi}{10} = \frac{\pi r^2}{20} \][/tex]

Now, the area [tex]\(A_{\text{circle}}\)[/tex] of the entire circle is:
[tex]\[ A_{\text{circle}} = \pi r^2 \][/tex]

Finally, the ratio of the area of the sector [tex]\(AOB\)[/tex] to the area of the whole circle is:
[tex]\[ \text{Ratio} = \frac{A_{\text{sector}}}{A_{\text{circle}}} = \frac{\frac{\pi r^2}{20}}{\pi r^2} = \frac{1}{20} \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{\frac{1}{20}} \][/tex]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.