Respuesta :
a.
We are given the following information
arc length, S = 105.75 cm
1/360 of the circumference is 0.75 cm long
and, we need to find the measure of the angle in degrees
First, let's calculate the value of the circumference, C
[tex]\begin{gathered} \frac{1}{360}\cdot C=0.75 \\ \Rightarrow C=0.75\cdot360=270 \end{gathered}[/tex]So, the circumference of the circle is 270 cm
Now, we can use the formula of the circumference to calculate the radius, r
[tex]\begin{gathered} C=2\pi r \\ r=\frac{C}{2\pi}=\frac{270}{2\pi}=\frac{135}{\pi}\cong42.97 \end{gathered}[/tex]So, the radiu of the circle is 135/pi or about 42.97 cm
Finally, let's use the arc length formula to calculate the angle, Θ
[tex]\begin{gathered} s=r\theta \\ \theta=\frac{s}{r}=\frac{105.75}{\frac{135}{\pi}}=\frac{105.75\cdot\pi}{135}\cong2.46\text{rad} \\ \theta=360\cdot\frac{s}{2\pi r}=180\cdot\frac{105.75}{\pi\cdot\frac{135}{\pi}}=\frac{180}{135}\cdot105.75=141\degree \end{gathered}[/tex]Thus, the angle is 141°
b.
we are given this information:
circumference, C = 414 cm
arc length, s = 259.9 cm
and, we need to find the measure of the angle in degrees
Now, let's use the following formulas
[tex]\begin{gathered} \theta=360\cdot\frac{s}{2\pi r} \\ r=\frac{C}{2\pi} \end{gathered}[/tex]let's calculare r first
[tex]r=\frac{414}{2\pi}=\frac{207}{\pi}[/tex]then, the angle is 225.22°
[tex]\theta=360\cdot\frac{259.9}{2\pi\cdot\frac{414}{2\pi}}=360\cdot\frac{259}{414}=226\degree[/tex]