The problem asks us to find the area of the shaded region and the length of the arc $AB$ of a circle with center $O$ and radius 6 yards, given that the measure of angle $AOB$ is 110 degrees. We need to provide exact answers in terms of $\pi$ and include the correct units.

1. Problem Description

The problem asks us to find the area of the shaded region and the length of the arc $AB$ of a circle with center $O$ and radius 6 yards, given that the measure of angle $AOB$ is 110 degrees. We need to provide exact answers in terms of $\pi$ and include the correct units.

2. Solution Steps

First, we find the area of the shaded region. The area of the entire circle is given by the formula:

$Area_{circle} = \pi r^2$

where $r$ is the radius. In this case, $r = 6$ yd, so

$Area_{circle} = \pi (6)^2 = 36\pi$ square yards.

The shaded region is a sector of the circle with a central angle of $110^\circ$. The fraction of the circle that the sector represents is $\frac{110}{360} = \frac{11}{36}$.

Therefore, the area of the shaded region is:

$Area_{sector} = \frac{11}{36} \times Area_{circle} = \frac{11}{36} \times 36\pi = 11\pi$ square yards.

Next, we find the length of the arc $AB$. The circumference of the entire circle is given by the formula:

$Circumference = 2\pi r$

In this case, $r = 6$ yd, so

$Circumference = 2\pi (6) = 12\pi$ yards.

The arc $AB$ is a fraction of the circumference, corresponding to the same fraction as the area of the sector, which is $\frac{110}{360} = \frac{11}{36}$.

Therefore, the length of the arc $AB$ is:

$Arc \, Length = \frac{11}{36} \times Circumference = \frac{11}{36} \times 12\pi = \frac{11 \times 12}{36}\pi = \frac{132}{36}\pi = \frac{11}{3}\pi$ yards.

3. Final Answer

Area of shaded region: $11\pi$ square yards

Length of arc $AB$: $\frac{11}{3}\pi$ yards