To find the angular speed [tex]\(\omega\)[/tex] in radians per second, given that the central angle [tex]\(\theta\)[/tex] is 288 degrees and the time [tex]\(t\)[/tex] is 4 seconds, follow these steps:
1. Convert the central angle from degrees to radians:
- The formula to convert degrees to radians is:
[tex]\[
\text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right)
\][/tex]
- Plugging in the value [tex]\(\theta = 288^{\circ}\)[/tex] into this formula, we get:
[tex]\[
\theta_{\text{radians}} = 288^{\circ} \times \left(\frac{\pi}{180}\right) = 288 \times \frac{\pi}{180} = 1.6\pi
\][/tex]
- After computing the numerical value, we have:
[tex]\[
\theta_{\text{radians}} \approx 5.026548245743669
\][/tex]
2. Calculate the angular speed [tex]\(\omega\)[/tex]:
- The formula to find angular speed is:
[tex]\[
\omega = \frac{\theta}{t}
\][/tex]
- Using [tex]\(\theta = 5.026548245743669 \text{ radians}\)[/tex] and [tex]\(t = 4 \text{ seconds}\)[/tex], we get:
[tex]\[
\omega = \frac{5.026548245743669}{4} = 1.2566370614359172 \text{ radians/second}
\][/tex]
Thus, the angular speed [tex]\(\omega\)[/tex] is [tex]\(\boxed{1.2566370614359172}\)[/tex] radians/second.
