Question
a) Find the length of an arc with radius 10m that subtends a central angle of 30°.
b) A central angle in a circle of radius 4m is subtends by an arc of length 6m. Find the measure of angle in radians.
c) find an area of a sector of a circle with central angle of 60° if the radius of the circle is 3m
b) A central angle in a circle of radius 4m is subtends by an arc of length 6m. Find the measure of angle in radians.
c) find an area of a sector of a circle with central angle of 60° if the radius of the circle is 3m
Answers
GPT-4o mini
Let's solve each part step by step.
### a) Length of an arc
The formula for the length of an arc \( L \) is given by:
\[
L = r \theta
\]
where:
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.
First, we need to convert degrees to radians. The conversion from degrees to radians is done using the formula:
\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\]
For a central angle of \( 30^\circ \):
\[
\theta = 30^\circ \times \frac{\pi}{180} = \frac{\pi}{6} \text{ radians}
\]
Now, substituting the values into the arc length formula:
\[
L = 10 \, \text{m} \times \frac{\pi}{6} = \frac{10\pi}{6} = \frac{5\pi}{3} \, \text{m}
\]
So, the length of the arc is approximately:
\[
L \approx 5.24 \, \text{m}
\]
### b) Measure of the angle in radians
To find the measure of the angle \( \theta \) in radians when given the arc length \( L \) and the radius \( r \), we use the formula:
\[
\theta = \frac{L}{r}
\]
Given:
- \( L = 6 \, \text{m} \)
- \( r = 4 \, \text{m} \)
Substituting the values:
\[
\theta = \frac{6}{4} = \frac{3}{2} \text{ radians}
\]
### c) Area of a sector
The formula for the area \( A \) of a sector is given by:
\[
A = \frac{1}{2} r^2 \theta
\]
Again, \( \theta \) must be in radians. For a central angle of \( 60^\circ \):
\[
\theta = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}
\]
Using the radius \( r = 3 \, \text{m} \):
\[
A = \frac{1}{2} \times 3^2 \times \frac{\pi}{3}
= \frac{1}{2} \times 9 \times \frac{\pi}{3}
= \frac{9\pi}{6}
= \frac{3\pi}{2} \, \text{m}^2
\]
So, the area of the sector is approximately:
\[
A \approx 4.71 \, \text{m}^2
\]
### Summary
a) Length of the arc: \( \frac{5\pi}{3} \, \text{m} \) or approximately \( 5.24 \, \text{m} \)
b) Measure of the angle: \( \frac{3}{2} \, \text{radians} \)
c) Area of the sector: \( \frac{3\pi}{2} \, \text{m}^2 \) or approximately \( 4.71 \, \text{m}^2 \)
### a) Length of an arc
The formula for the length of an arc \( L \) is given by:
\[
L = r \theta
\]
where:
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.
First, we need to convert degrees to radians. The conversion from degrees to radians is done using the formula:
\[
\text{radians} = \text{degrees} \times \frac{\pi}{180}
\]
For a central angle of \( 30^\circ \):
\[
\theta = 30^\circ \times \frac{\pi}{180} = \frac{\pi}{6} \text{ radians}
\]
Now, substituting the values into the arc length formula:
\[
L = 10 \, \text{m} \times \frac{\pi}{6} = \frac{10\pi}{6} = \frac{5\pi}{3} \, \text{m}
\]
So, the length of the arc is approximately:
\[
L \approx 5.24 \, \text{m}
\]
### b) Measure of the angle in radians
To find the measure of the angle \( \theta \) in radians when given the arc length \( L \) and the radius \( r \), we use the formula:
\[
\theta = \frac{L}{r}
\]
Given:
- \( L = 6 \, \text{m} \)
- \( r = 4 \, \text{m} \)
Substituting the values:
\[
\theta = \frac{6}{4} = \frac{3}{2} \text{ radians}
\]
### c) Area of a sector
The formula for the area \( A \) of a sector is given by:
\[
A = \frac{1}{2} r^2 \theta
\]
Again, \( \theta \) must be in radians. For a central angle of \( 60^\circ \):
\[
\theta = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}
\]
Using the radius \( r = 3 \, \text{m} \):
\[
A = \frac{1}{2} \times 3^2 \times \frac{\pi}{3}
= \frac{1}{2} \times 9 \times \frac{\pi}{3}
= \frac{9\pi}{6}
= \frac{3\pi}{2} \, \text{m}^2
\]
So, the area of the sector is approximately:
\[
A \approx 4.71 \, \text{m}^2
\]
### Summary
a) Length of the arc: \( \frac{5\pi}{3} \, \text{m} \) or approximately \( 5.24 \, \text{m} \)
b) Measure of the angle: \( \frac{3}{2} \, \text{radians} \)
c) Area of the sector: \( \frac{3\pi}{2} \, \text{m}^2 \) or approximately \( 4.71 \, \text{m}^2 \)