Respuesta :
Answer:
D. [tex]\frac{27\pi}{2}[/tex] inches.
Step-by-step explanation:
We have been given that the circle with center O has a minor arc BSA with a length of [tex]3\pi^{2}[/tex] inches. The central angle is 40°.
To find the circumference of circle we will use formula:
[tex]\frac{\text{Central angle}}{2\pi}=\frac{\text{Arc length}}{2\pi r}[/tex], where [tex]2\pi[/tex]= measure of 360 degrees in radians and [tex]2\pi r[/tex]= circumference of circle.
Let us convert measure of central angle into radians.
[tex]40^{o}=\frac{40*\pi}{180} =\frac{2\pi}{9}[/tex]
Upon substituting our given value in the formula we will get,
[tex]\frac{\frac{2\pi}{9}}{2\pi}=\frac{\frac{3\pi}{2}}{2\pi r}[/tex]
[tex]\frac{2\pi}{18\pi}=\frac{3\pi}{4\pi r}[/tex]
[tex]\frac{1}{9}=\frac{3}{4r}[/tex]
Cross multiplying we will get,
[tex]4r=27[/tex]
[tex]r=\frac{27}{4}[/tex]
Hence, the radius of our circle is 27/4 inches.
Since the circumference of circle is [tex]2\pi r[/tex]. Upon substituting [tex]r=\frac{27}{4}[/tex] we will get,
[tex]2\pi r=2\pi* \frac{27}{4}=\frac{27\pi}{2}[/tex]
Therefore, circumference of our given circle will be [tex]\frac{27\pi}{2}[/tex] inches and option D is the correct choice.
