To find the area of a quadrant of a circle given that the perimeter of the quadrant is 50 cm, we proceed step by step as follows:
1. Understand the Components of the Perimeter:
- A quadrant of a circle consists of a quarter of the circular arc plus two radii of the circle.
- The formula for the perimeter of a quadrant is:
[tex]\[
\text{Perimeter} = \frac{1}{4} \times 2 \pi r + 2r = \frac{1}{2} \pi r + 2r
\][/tex]
Here, [tex]\( r \)[/tex] is the radius of the full circle.
2. Set Up the Equation:
- We are given that the perimeter of the quadrant is 50 cm.
- Therefore, the equation becomes:
[tex]\[
\frac{1}{2} \pi r + 2r = 50
\][/tex]
3. Solve for the Radius [tex]\( r \)[/tex]:
- We need to solve the equation [tex]\(\frac{1}{2} \pi r + 2r = 50\)[/tex] for [tex]\( r \)[/tex].
By isolating [tex]\( r \)[/tex], it is found that:
[tex]\[
r \approx 14.0056 \text{ cm}
\][/tex]
4. Calculate the Area of the Quadrant:
- The area of a full circle is given by the formula [tex]\( \pi r^2 \)[/tex].
- Therefore, the area of a quadrant (which is one-fourth of the circle) is:
[tex]\[
\text{Area of a Quadrant} = \frac{1}{4} \pi r^2
\][/tex]
- Substituting the radius [tex]\( r \approx 14 cm \)[/tex] into this formula:
[tex]\[
\text{Area of a Quadrant} \approx \frac{1}{4} \pi (14.0056)^2
\][/tex]
- This gives us the area of the quadrant:
[tex]\[
\text{Area of a Quadrant} \approx 153.983 \text{ cm}^2
\][/tex]
In conclusion, the radius of the circle is approximately [tex]\( 14.0056 \)[/tex] cm, and the area of the quadrant is approximately [tex]\( 153.983 \)[/tex] square centimeters.
