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 it 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 combines is about ft².

Explanation:

Step1: Recall hexagon side - diagonal relation

For a regular hexagon, if the length of the long - diagonal (distance from one vertex to the opposite vertex) is \(d\) and the side length is \(a\), then \(d = 2a\). Given \(d=30\) ft, so \(a=\frac{30}{2}=15\) ft.

Step2: Use hexagon area formula

The area formula for a regular hexagon is \(A=\frac{3\sqrt{3}}{2}a^{2}\). Substitute \(a = 15\) ft into the formula. \(A=\frac{3\sqrt{3}}{2}\times15^{2}=\frac{3\sqrt{3}}{2}\times225\approx\frac{3\times1.732}{2}\times225 = 3\times0.866\times225= 3\times194.85 = 584.55\approx585\) \(ft^{2}\). Since the entrance and the seating section together form a hexagon, the combined area is about \(585\) \(ft^{2}\). The area of the seating section and the entrance separately are not given enough information to calculate individually in a non - assumed way. But if we assume the hexagon is regular, we have found the combined area.

Answer:

Each side of the hexagon is about 15 ft long. The area of the seating section is about 585 \(ft^{2}\) (combined with entrance). The area of the entrance is about (not enough info to determine separately). The area of the entrance and seating section combined is about 585 \(ft^{2}\)