Question

select the correct answer from each drop - down menu. the hexagonal seating section in an auditorium has a small entrance at the front. a walkway leading from the entrance to the front of the stage is 30 ft long. what is the approximate area of the entrance and seating section combined? each side of the hexagon is about ft long. the area of the seating section is about ft². the area of the entrance is about ft². the area of the entrance and seating section combined is about ft²

Explanation:

Step1: Recall hexagon side - diagonal relation

For a regular hexagon, if the length of the diagonal (from one vertex to the opposite vertex) is \(d = 30\) ft, and the side - length \(s\) of a regular hexagon is related to the diagonal by \(d = 2s\). So, \(s=\frac{d}{2}\).
\[s=\frac{30}{2}=15\] ft.

Step2: Calculate the area of a regular hexagon

The area formula for a regular hexagon is \(A=\frac{3\sqrt{3}}{2}s^{2}\). Substitute \(s = 15\) ft into the formula.
\[A=\frac{3\sqrt{3}}{2}\times15^{2}=\frac{3\sqrt{3}}{2}\times225\approx\frac{3\times1.732}{2}\times225 = 3\times0.866\times225=584.85\approx585\] \(ft^{2}\).
Assume the entrance area is small and negligible compared to the hexagon - shaped seating area. So the combined area is approximately the area of the hexagon.

Answer:

Each side of the hexagon is about 15 ft long. The area of the seating section is about 585 \(ft^{2}\). The area of the entrance is about 0 \(ft^{2}\) (negligible). The area of the entrance and seating section combined is about 585 \(ft^{2}\).