To solve this question, we'll break it down into smaller parts and understand each step involving geometrical relationships and formulas. Let's begin:
1. Height of the Pyramid:
- The length of the base edge of the pyramid is denoted as [tex]\( x \)[/tex].
- We're given that the height of the pyramid is 3 times the length of the base edge.
- Therefore, the height of the pyramid can be represented as:
[tex]\[
3x
\][/tex]
2. Area of an Equilateral Triangle:
- The area of an equilateral triangle with side length [tex]\( x \)[/tex] can be calculated using the formula:
[tex]\[
\frac{x^2 \sqrt{3}}{4}
\][/tex]
- Thus, the area of an equilateral triangle with side length [tex]\( x \)[/tex] is:
[tex]\[
\frac{x^2 \sqrt{3}}{4} \text{ square units}
\][/tex]
3. Area of the Hexagon Base:
- A regular hexagon can be divided into 6 equilateral triangles.
- Therefore, the area of the hexagon base will be 6 times the area of one equilateral triangle.
- Given the area of one equilateral triangle is [tex]\( \frac{x^2 \sqrt{3}}{4} \)[/tex], the area of the hexagon base is:
[tex]\[
6 \times \frac{x^2 \sqrt{3}}{4} = \frac{3 \sqrt{3}}{2}x^2
\][/tex]
4. Volume of the Pyramid:
- The volume of a pyramid is calculated using the formula:
[tex]\[
\text{Volume of a pyramid} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\][/tex]
- Plugging in the base area [tex]\( \frac{3 \sqrt{3}}{2}x^2 \)[/tex] and height [tex]\( 3x \)[/tex]:
[tex]\[
\text{Volume} = \frac{1}{3} \times \frac{3 \sqrt{3}}{2}x^2 \times 3x
\][/tex]
- Simplifying the equation:
[tex]\[
= \frac{1}{3} \times \frac{9 \sqrt{3}}{2} x^3
\][/tex]
[tex]\[
= \frac{9 \sqrt{3}}{6} x^3
\][/tex]
[tex]\[
= 1.5 \sqrt{3} x^3
\][/tex]
- So, the volume of the pyramid is:
[tex]\[
1.5 \sqrt{3} x^3 \text{ cubic units}
\][/tex]
Thus, the final answers for the blanks are:
- The height of the pyramid can be represented as:
[tex]\[
3x
\][/tex]
- The [tex]\(\square\)[/tex] of an equilateral triangle with length [tex]\( x \)[/tex] is [tex]\(\frac{x^2 \sqrt{3}}{4}\)[/tex] units[tex]\(^2\)[/tex].
- The area of the hexagon base is [tex]\( 6 \)[/tex] times the area of the equilateral triangle.
- The volume of the pyramid is [tex]\( 1.5 \sqrt{3} x^3 \)[/tex] units[tex]\(^3\)[/tex].
