Given: A circle with a diameter of 19 ft and a minor sector with a central angle of 123 degrees.
Required: To find out the area of a major sector.
Explanation: The area of the sector is given by the formula
[tex]A=\frac{\theta}{360}\times\pi r^2[/tex]
where,
[tex]\theta\text{ is the central angle and }r\text{ is the radius of the circle. }[/tex]
Now, the angle of the major sector (or the shaded region) can be found as follows,
[tex]\begin{gathered} \text{ For the major sector, }\theta=360\degree-123\degree \\ =237\degree \\ r=\frac{19}{2}\text{ ft} \end{gathered}[/tex]
Now, the Area of the shaded region is
[tex]A=\frac{237}{360}\times\frac{22}{7}\times(\frac{19}{2})^2[/tex]
Which on simplification gives the result,
[tex]A=186.73\text{ feet}^2[/tex]
Final Answer: The area of the shaded region is 186.73 square feet.
