1) Given the angles below transform into degrees or radians as needed, round to two decimal points if necessary.
[tex]\begin{gathered} 4\pi \\ 720\degree \\ 10\degree \\ \frac{5}{8}\pi \end{gathered}[/tex]
In general, 2pi=360°; then, 4pi radians refer to 2 revolutions around a circumference, as shown in the diagram below
Therefore, 4pi=0°=360°.
Similarly,
[tex]\begin{gathered} \frac{360\degree}{2\pi}=\frac{720\degree}{x} \\ \Rightarrow x=2\pi(\frac{720}{360})=2\pi *2=4\pi=0 \end{gathered}[/tex]
720°=0 radians
[tex]\begin{gathered} \frac{2\pi}{360\degree}=\frac{x}{10\degree} \\ \Rightarrow x=2\pi(\frac{10}{360})=\frac{\pi}{18} \end{gathered}[/tex]
10°=pi/18 radians.
[tex]\begin{gathered} \frac{x}{\frac{5}{8}\pi}=\frac{360\degree}{2\pi} \\ \Rightarrow x=360\degree(\frac{\frac{5}{8}\pi}{2\pi})=360\degree(\frac{5}{16})=112.5 \end{gathered}[/tex]
5pi/8=112.5°.
2) Given that the area of a circle is 30pi and an arc length on it is pi/2; find the central angle generated by such arc in radians, round to three decimal places if needed.
The area of a circle is given by
[tex]\begin{gathered} A=\pi r^2 \\ r\rightarrow radius \end{gathered}[/tex]
Then, in our case,
[tex]30\pi=r^2\pi\Rightarrow r=\sqrt{30}[/tex]
On the other hand, an arc length is given by the formula below
[tex]arc=r\theta[/tex]
In our case,
[tex]\begin{gathered} arc=\frac{\pi}{2} \\ \Rightarrow\frac{\pi}{2}=\sqrt{30}\theta \\ \Rightarrow\theta=\frac{\pi}{2\sqrt{30}} \\ \Rightarrow\theta\approx0.287\text{ radians} \end{gathered}[/tex]
The answer is 0.287 radians.
