Let's break down the problem step by step to understand the mathematical concepts involved.
1. Height of the pyramid:
The height of the pyramid is given to be 3 times the length of the base edge [tex]\( x \)[/tex].
[tex]\[
\text{Height} = 3x
\][/tex]
2. Area of one equilateral triangle in the hexagon base:
The hexagon can be divided into six equilateral triangles, each with side length [tex]\( x \)[/tex].
The formula for the area of an equilateral triangle with side length [tex]\( x \)[/tex] is:
[tex]\[
\text{Area of one equilateral triangle} = \frac{\sqrt{3}}{4} x^2
\][/tex]
This is given in the problem as [tex]\(\frac{x^2 \sqrt{3}}{4}\)[/tex] units[tex]\(^2\)[/tex].
3. Area of the hexagon base:
Since the hexagon is composed of six equilateral triangles, the area of the hexagon base will be six times the area of one equilateral triangle.
[tex]\[
\text{Area of the hexagon base} = 6 \times \left(\frac{\sqrt{3}}{4} x^2\right) = \frac{6\sqrt{3}}{4} x^2 = \frac{3\sqrt{3}}{2} x^2
\][/tex]
4. Volume of the pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is given by:
[tex]\[
V = \frac{1}{3} \times \text{Area of the base} \times \text{Height}
\][/tex]
Here, substituting the area of the base and the height:
[tex]\[
V = \frac{1}{3} \times \left(\frac{3\sqrt{3}}{2} x^2\right) \times 3x
\][/tex]
Simplifying this:
[tex]\[
V = \frac{1}{3} \times \left(\frac{3\sqrt{3}}{2} x^2 \times 3x\right) = \frac{1}{3} \times \frac{9\sqrt{3}}{2} x^3 = \frac{9\sqrt{3}}{6} x^3 = \frac{3\sqrt{3}}{2} x^3
\][/tex]
Therefore, the detailed step-by-step solution is:
1. Height of the pyramid:
[tex]\[
3x
\][/tex]
2. Area of one equilateral triangle:
[tex]\[
\frac{\sqrt{3}}{4} x^2
\][/tex]
3. Area of the hexagon base:
[tex]\[
\frac{3\sqrt{3}}{2} x^2
\][/tex]
4. Volume of the pyramid:
[tex]\[
\frac{3\sqrt{3}}{2} x^3
\][/tex]
So, the area of each equilateral triangle is [tex]\(\frac{\sqrt{3}}{4} x^2\)[/tex] units[tex]\(^2\)[/tex], the area of the hexagon base is [tex]\(\frac{3\sqrt{3}}{2} x^2\)[/tex] units[tex]\(^2\)[/tex], and the volume of the pyramid is [tex]\(\frac{3\sqrt{3}}{2} x^3\)[/tex] units[tex]\(^3\)[/tex].
