Find the area of a sector of a circle having radius $r$ and central angle $\theta$.
\[
r=10.5 \mathrm{~cm}, \theta=82^{\circ}
\]
The area is approximately $\square \mathrm{cm}^{2}$.
(Do not round until the final answer. Then round to the nearest tenth as needed.)

Step 1 ：We are given the radius \(r = 10.5 \, \text{cm}\) and the central angle \(\theta = 82^\circ\).

Step 2 ：The area of a sector of a circle can be calculated using the formula \(A = \frac{1}{2} r^2 \theta\), where \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.

Step 3 ：However, in this problem, the central angle is given in degrees. Therefore, we need to convert the central angle from degrees to radians before we can use the formula. The conversion from degrees to radians is done by multiplying the angle in degrees by \(\frac{\pi}{180}\).

Step 4 ：After converting the central angle to radians, we get \(\theta = 1.4311699866353502\) radians.

Step 5 ：We can now substitute the values of \(r\) and \(\theta\) into the formula to find the area of the sector. Doing so, we get \(A = 78.89324551327368 \, \text{cm}^2\).

Step 6 ：Rounding to the nearest tenth as needed, we get \(A = 78.9 \, \text{cm}^2\).

Step 7 ：Final Answer: The area of the sector of the circle is approximately \(\boxed{78.9 \, \text{cm}^2}\).