The formula to find the arc length of a circle is:
[tex]\begin{gathered} \text{arc length }=\frac{\theta}{360\degree}\cdot2\pi r \\ \text{ Where} \\ \theta\text{ is the measure of the central angle} \\ r\text{ is the radius of the circle } \end{gathered}[/tex]
In this case, we have:
[tex]\begin{gathered} \theta=49\degree \\ r=10in \\ \text{arc length }=\frac{\theta}{360\degree}\cdot2\pi r \\ \text{arc length }=\frac{49\degree}{360\degree}\cdot2\pi(10in) \\ \text{arc length }=\frac{49}{360}\cdot2\pi\cdot10in \\ \text{arc length }=\frac{49\cdot2\cdot10}{360}\pi in \\ \text{arc length }=\frac{49\cdot2\cdot10}{18\cdot2\cdot10}\pi in \\ \text{arc length }=\frac{49}{18}\pi in \\ \text{ or} \\ \text{arc length }\approx2.72\pi in\Rightarrow\text{ The symbol }\approx\text{ is read 'approximately'.} \end{gathered}[/tex]
Therefore, the length of the curved boundary of the piece of the cardboard Ramsay cuts out is
[tex]\begin{gathered} \frac{49}{18}\pi\text{ inches} \\ \text{ or} \\ 2.72\pi\text{ inches} \end{gathered}[/tex]
