Let's follow the details step-by-step to determine the required expressions for the volume of the pyramid:
1. Expressing the Height of the Pyramid:
The height of the pyramid [tex]\( h \)[/tex] is given to be 3 times the length of the base edge [tex]\( x \)[/tex]. Therefore,
[tex]\[
h = 3x
\][/tex]
Thus, the height of the pyramid can be represented as:
[tex]\[
3x
\][/tex]
2. Area of an Equilateral Triangle:
The area [tex]\( A_{\triangle} \)[/tex] of an equilateral triangle with side length [tex]\( x \)[/tex] is given by:
[tex]\[
A_{\triangle} = \frac{x^2 \sqrt{3}}{4}
\][/tex]
Thus, the area of the equilateral triangle with length [tex]\( x \)[/tex] is:
[tex]\[
\frac{x^2 \sqrt{3}}{4} \text{ units}^2
\][/tex]
3. Area of the Hexagon Base:
A regular hexagon can be divided into 6 equilateral triangles. Therefore, the area of the hexagon base [tex]\( A_{\text{hex}} \)[/tex] is 6 times the area of one equilateral triangle:
[tex]\[
A_{\text{hex}} = 6 \times \frac{x^2 \sqrt{3}}{4} = \frac{6x^2 \sqrt{3}}{4} = \frac{3x^2 \sqrt{3}}{2}
\][/tex]
Thus, the area of the hexagon base is:
[tex]\[
3 \times \frac{x^2 \sqrt{3}}{2}
\][/tex]
times the area of the equilateral triangle.
4. Volume of the Pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is calculated using the formula:
[tex]\[
V = \frac{1}{3} \times \text{Base area} \times \text{Height}
\][/tex]
Substituting the respective values, we have:
[tex]\[
V = \frac{1}{3} \times \frac{3x^2 \sqrt{3}}{2} \times 3x = \frac{1}{3} \times \frac{9x^3 \sqrt{3}}{2} = \frac{9x^3 \sqrt{3}}{6} = \frac{3x^3 \sqrt{3}}{2} = 1.5 \cdot \left( x^3 \sqrt{3} \right)
\][/tex]
Thus, the volume of the pyramid is:
[tex]\[
1.5 \cdot x^3 \sqrt{3} \text{ units}^3
\][/tex]
So, filling in the blanks, the detailed solution is:
- The height of the pyramid can be represented as [tex]\( 3x \)[/tex].
- The area of an equilateral triangle with length [tex]\( x \)[/tex] is [tex]\( \frac{x^2 \sqrt{3}}{4} \text{ units}^2 \)[/tex].
- The area of the hexagon base is [tex]\( 3 \)[/tex] times the area of the equilateral triangle.
- The volume of the pyramid is [tex]\( 1.5 \cdot x^3 \sqrt{3} \text{ units}^3 \)[/tex].
