Question

a right pyramid with a regular hexagon base has a base edge length of 4 ft and a height of 10 ft.
\\(\overline{wy}\\) represents the
if the area of an equ le with sides of 4 ft is 4 \\(\sqrt{3}\\) square feet, the the regular hexagon base is \\(\sqrt{3}\\) squ
the volume is
(options for dropdown: apex, height, lateral side)

Explanation:

Part 1: Identifying \(\overline{WY}\)

In a right pyramid, the segment from the center of the base (\(W\)) to the apex (\(Y\)) is the height of the pyramid. So \(\overline{WY}\) represents the height of the pyramid.

Part 2: Area of the Regular Hexagon Base

A regular hexagon can be divided into 6 equilateral triangles, each with side length equal to the side length of the hexagon (4 ft here).
Given the area of one equilateral triangle with side 4 ft is \(4\sqrt{3}\) sq ft.
Number of such triangles in the hexagon = 6.
So area of the hexagon base \(A = 6\times4\sqrt{3}=24\sqrt{3}\) square feet.

Part 3: Volume of the Pyramid

The formula for the volume \(V\) of a pyramid is \(V=\frac{1}{3}Bh\), where \(B\) is the area of the base and \(h\) is the height of the pyramid.
Here, \(B = 24\sqrt{3}\) sq ft and \(h = 10\) ft.
Substitute into the formula:
\(V=\frac{1}{3}\times24\sqrt{3}\times10\)
Simplify:
\(\frac{1}{3}\times24\sqrt{3}\times10 = 8\sqrt{3}\times10 = 80\sqrt{3}\) cubic feet.

Final Answers:

• \(\overline{WY}\) represents the \(\boldsymbol{\text{height}}\).
• Area of the regular hexagon base: \(\boldsymbol{24}\sqrt{3}\) square feet.
• Volume of the pyramid: \(\boldsymbol{80\sqrt{3}}\) cubic feet.

Answer:

Part 1: Identifying \(\overline{WY}\)

In a right pyramid, the segment from the center of the base (\(W\)) to the apex (\(Y\)) is the height of the pyramid. So \(\overline{WY}\) represents the height of the pyramid.

Part 2: Area of the Regular Hexagon Base

A regular hexagon can be divided into 6 equilateral triangles, each with side length equal to the side length of the hexagon (4 ft here).
Given the area of one equilateral triangle with side 4 ft is \(4\sqrt{3}\) sq ft.
Number of such triangles in the hexagon = 6.
So area of the hexagon base \(A = 6\times4\sqrt{3}=24\sqrt{3}\) square feet.

Part 3: Volume of the Pyramid

The formula for the volume \(V\) of a pyramid is \(V=\frac{1}{3}Bh\), where \(B\) is the area of the base and \(h\) is the height of the pyramid.
Here, \(B = 24\sqrt{3}\) sq ft and \(h = 10\) ft.
Substitute into the formula:
\(V=\frac{1}{3}\times24\sqrt{3}\times10\)
Simplify:
\(\frac{1}{3}\times24\sqrt{3}\times10 = 8\sqrt{3}\times10 = 80\sqrt{3}\) cubic feet.

Final Answers:

• \(\overline{WY}\) represents the \(\boldsymbol{\text{height}}\).
• Area of the regular hexagon base: \(\boldsymbol{24}\sqrt{3}\) square feet.
• Volume of the pyramid: \(\boldsymbol{80\sqrt{3}}\) cubic feet.