To determine the area of the circular opening of the duct, we need to follow these steps:
1. Find the radius of the duct:
The diameter of the duct is given as 16 inches. The radius (r) is half of the diameter.
[tex]\[
\text{radius} = \frac{\text{diameter}}{2} = \frac{16}{2} = 8 \text{ inches}
\][/tex]
2. Calculate the area of the circular opening:
The formula for the area (A) of a circle is given by:
[tex]\[
A = \pi r^2
\][/tex]
Substituting the value of the radius:
[tex]\[
A = \pi \times (8)^2
\][/tex]
[tex]\[
A = \pi \times 64
\][/tex]
3. Use the value of [tex]\( \pi \)[/tex]:
For the calculation, we use the approximation [tex]\( \pi \approx 3.14159 \)[/tex].
[tex]\[
A \approx 3.14159 \times 64 \approx 201.06192982974676
\][/tex]
4. Round the result to the nearest whole number:
The area calculated is approximately 201.06192982974676 square inches. Rounding this to the nearest whole number gives:
[tex]\[
A \approx 201 \text{ square inches}
\][/tex]
Therefore, the area of the circular opening of the duct, rounded to the nearest whole number, is 201 square inches.
