Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To find the exact length of the arc created by a central angle [tex]\(\theta = 60^\circ\)[/tex] in a circle with a radius of [tex]\(r = 132\)[/tex] cm, we can follow these steps:
1. Convert the central angle from degrees to radians:
A central angle in degrees can be converted to radians by using the formula:
[tex]\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \][/tex]
For [tex]\(\theta = 60^\circ\)[/tex]:
[tex]\[ \theta_{\text{radians}} = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \][/tex]
2. Calculate the arc length:
The formula for the arc length [tex]\(s\)[/tex] of a circle is given by:
[tex]\[ s = r \times \theta_{\text{radians}} \][/tex]
Substituting the known values [tex]\(r = 132\)[/tex] cm and [tex]\(\theta_{\text{radians}} = \frac{\pi}{3}\)[/tex]:
[tex]\[ s = 132 \times \frac{\pi}{3} \][/tex]
3. Simplify the expression:
Simplify the multiplication:
[tex]\[ s = \frac{132 \pi}{3} = 44 \pi \][/tex]
Therefore, the exact length of the arc is:
[tex]\[ s = 44 \pi \text{ cm} \][/tex]
1. Convert the central angle from degrees to radians:
A central angle in degrees can be converted to radians by using the formula:
[tex]\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \][/tex]
For [tex]\(\theta = 60^\circ\)[/tex]:
[tex]\[ \theta_{\text{radians}} = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \][/tex]
2. Calculate the arc length:
The formula for the arc length [tex]\(s\)[/tex] of a circle is given by:
[tex]\[ s = r \times \theta_{\text{radians}} \][/tex]
Substituting the known values [tex]\(r = 132\)[/tex] cm and [tex]\(\theta_{\text{radians}} = \frac{\pi}{3}\)[/tex]:
[tex]\[ s = 132 \times \frac{\pi}{3} \][/tex]
3. Simplify the expression:
Simplify the multiplication:
[tex]\[ s = \frac{132 \pi}{3} = 44 \pi \][/tex]
Therefore, the exact length of the arc is:
[tex]\[ s = 44 \pi \text{ cm} \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.