Let's solve the problem step-by-step:
1. Understand the given information:
- The angle between the arms of the divider is 15 degrees.
- The length of each arm (radius of the sector) is 8.4 cm.
- We are to use [tex]\(\pi = \frac{22}{7}\)[/tex].
2. Convert the angle from degrees to radians:
Generally, to convert an angle from degrees to radians, we use the conversion factor:
[tex]\[
\text{angle in radians} = \text{angle in degrees} \times \left( \frac{\pi}{180} \right)
\][/tex]
Here, [tex]\(\pi = \frac{22}{7}\)[/tex] and the angle is 15 degrees. Plugging in these values:
[tex]\[
\text{angle in radians} = 15 \times \left( \frac{ \frac{22}{7} }{ 180 } \right) = 15 \times \frac{22}{ 1260 }
\][/tex]
Simplifying [tex]\( \frac{22}{1260} \)[/tex]:
[tex]\[
\frac{22}{1260} = \frac{1}{57.27}
\][/tex]
So,
[tex]\[
\text{angle in radians} = 15 \times \frac{1}{57.27} \approx 0.2619047619047619 \text{ radians}
\][/tex]
3. Calculate the area of the sector:
The formula to calculate the area of a sector when the angle is in radians is:
[tex]\[
\text{Area} = \frac{1}{2} \times r^2 \times \theta
\][/tex]
where [tex]\( r \)[/tex] is the radius, and [tex]\( \theta \)[/tex] is the angle in radians. Here, [tex]\( r = 8.4 \, \text{cm} \)[/tex] and [tex]\( \theta \approx 0.2619 \, \text{radians} \)[/tex]. Plugging in these values:
[tex]\[
\text{Area} = \frac{1}{2} \times (8.4)^2 \times 0.2619
\][/tex]
First, compute [tex]\( (8.4)^2 \)[/tex]:
[tex]\[
(8.4)^2 = 70.56
\][/tex]
Then, compute:
[tex]\[
\frac{1}{2} \times 70.56 \times 0.2619 \approx 9.24 \, \text{cm}^2
\][/tex]
4. Conclude the answer:
So, the area of the sector formed by the pair of divider with an arm length of 8.4 cm and an angle of 15 degrees is approximately 9.24 square centimeters.
