Let's break down the problem step-by-step:
1. Height Representation:
- The height of the pyramid is 3 times the length of its base edge, \(x\).
- Therefore, the height of the pyramid can be represented as \(3x\).
2. Area of the Equilateral Triangle:
- The area of an equilateral triangle with side length \(x\) is given by:
[tex]\[
\text{Area}_{\text{equilateral triangle}} = \frac{x^2 \sqrt{3}}{4}
\][/tex]
3. Area of the Hexagon Base:
- A regular hexagon can be divided into 6 equilateral triangles.
- Thus, the area of the hexagon base is 6 times the area of one equilateral triangle:
[tex]\[
\text{Area}_{\text{hexagon base}} = 6 \times \left(\frac{x^2 \sqrt{3}}{4}\right) = \frac{3 x^2 \sqrt{3}}{2}
\][/tex]
4. Volume of the Pyramid:
- The volume of a pyramid is calculated using the formula:
[tex]\[
\text{Volume}_{\text{pyramid}} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\][/tex]
- Plugging in the hexagon base area and the height, we get:
[tex]\[
\text{Volume}_{\text{pyramid}} = \frac{1}{3} \times \left(\frac{3 x^2 \sqrt{3}}{2}\right) \times (3x)
\][/tex]
- Simplifying the above expression:
[tex]\[
\text{Volume}_{\text{pyramid}} = \frac{3 x^2 \sqrt{3}}{2} \times x = \frac{3}{2} \times x^3 \times \sqrt{3} = 1.5 \cdot x^3 \cdot \sqrt{3}
\][/tex]
Now placing the results in the given question format:
The height of the pyramid can be represented as \(3x\).
The area of an equilateral triangle with length \(x\) is \(\frac{x^2 \sqrt{3}}{4}\) units\(^2\).
The area of the hexagon base is \(6\) times the area of the equilateral triangle.
The volume of the pyramid is [tex]\(1.5 \cdot x^3 \cdot \sqrt{3}\)[/tex] units[tex]\(^3\)[/tex].
