Question

I need help question

Answer 1

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

`\begin{gathered} (1)/(360)\cdot C=0.75 \\ \Rightarrow C=0.75\cdot360=270 \end{gathered}`

So, the circumference of the circle is 270 cm

Now, we can use the formula of the circumference to calculate the radius, r

`\begin{gathered} C=2\pi r \\ r=(C)/(2\pi)=(270)/(2\pi)=(135)/(\pi)\cong42.97 \end{gathered}`

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

`\begin{gathered} s=r\theta \\ \theta=(s)/(r)=(105.75)/((135)/(\pi))=(105.75\cdot\pi)/(135)\cong2.46\text{rad} \\ \theta=360\cdot(s)/(2\pi r)=180\cdot(105.75)/(\pi\cdot(135)/(\pi))=(180)/(135)\cdot105.75=141\degree \end{gathered}`

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

`\begin{gathered} \theta=360\cdot(s)/(2\pi r) \\ r=(C)/(2\pi) \end{gathered}`

let's calculare r first

`r=(414)/(2\pi)=(207)/(\pi)`

then, the angle is 225.22°

`\theta=360\cdot(259.9)/(2\pi\cdot(414)/(2\pi))=360\cdot(259)/(414)=226\degree`