Answer :
Sure! Let's solve this problem step-by-step:
1. Understand the problem: We have a circle with center [tex]\( O \)[/tex] and radius [tex]\( OA = 5 \)[/tex]. We need to find the area of the sector [tex]\( AOB \)[/tex]. The provided ratio of the central angle to the full circle is [tex]\( \frac{1}{4} \)[/tex].
2. Full circle and fraction details:
- The full circle in degrees is 360°, and in radians, it is [tex]\( 2\pi \)[/tex]. We are told to use the value [tex]\( \pi = 3.14 \)[/tex].
- Therefore, the full circle in radians is approximately [tex]\( 2 \times 3.14 = 6.28 \)[/tex].
3. Sector angle in radians:
- The sector's angle is [tex]\( \frac{1}{4} \)[/tex] of the full circle. So, we calculate it as:
[tex]\[
\text{Sector angle} = \frac{1}{4} \times 6.28 = 1.57
\][/tex]
4. Calculate the area of the sector:
- The formula for the area of a sector is:
[tex]\[
\text{Area of sector} = \left(\frac{\text{Sector angle in radians}}{\text{Angle of full circle in radians}}\right) \times \pi \times \text{radius}^2
\][/tex]
- Substitute the known values:
[tex]\[
\text{Area of sector} = \left(\frac{1.57}{6.28}\right) \times 3.14 \times 5^2
\][/tex]
- Calculate the expression:
[tex]\[
\text{Area of sector} = \frac{1.57 \times 3.14 \times 25}{6.28} = 19.625
\][/tex]
5. Select the closest answer:
- Among the given options, the closest value to 19.625 is 19.6 square units.
Thus, the answer is [tex]\( \boxed{19.6} \)[/tex] square units.
1. Understand the problem: We have a circle with center [tex]\( O \)[/tex] and radius [tex]\( OA = 5 \)[/tex]. We need to find the area of the sector [tex]\( AOB \)[/tex]. The provided ratio of the central angle to the full circle is [tex]\( \frac{1}{4} \)[/tex].
2. Full circle and fraction details:
- The full circle in degrees is 360°, and in radians, it is [tex]\( 2\pi \)[/tex]. We are told to use the value [tex]\( \pi = 3.14 \)[/tex].
- Therefore, the full circle in radians is approximately [tex]\( 2 \times 3.14 = 6.28 \)[/tex].
3. Sector angle in radians:
- The sector's angle is [tex]\( \frac{1}{4} \)[/tex] of the full circle. So, we calculate it as:
[tex]\[
\text{Sector angle} = \frac{1}{4} \times 6.28 = 1.57
\][/tex]
4. Calculate the area of the sector:
- The formula for the area of a sector is:
[tex]\[
\text{Area of sector} = \left(\frac{\text{Sector angle in radians}}{\text{Angle of full circle in radians}}\right) \times \pi \times \text{radius}^2
\][/tex]
- Substitute the known values:
[tex]\[
\text{Area of sector} = \left(\frac{1.57}{6.28}\right) \times 3.14 \times 5^2
\][/tex]
- Calculate the expression:
[tex]\[
\text{Area of sector} = \frac{1.57 \times 3.14 \times 25}{6.28} = 19.625
\][/tex]
5. Select the closest answer:
- Among the given options, the closest value to 19.625 is 19.6 square units.
Thus, the answer is [tex]\( \boxed{19.6} \)[/tex] square units.