Answer :
Sure, let's work through the problem step by step to find the area of the regular hexagon base and the volume of the right pyramid.
### Step 1: Calculate the Area of the Regular Hexagon Base
Given:
- The base edge length of the hexagon is [tex]\( 4 \)[/tex] feet.
- The area of an equilateral triangle with sides [tex]\( 4 \)[/tex] feet is [tex]\( 4 \sqrt{3} \)[/tex] square feet.
A regular hexagon can be divided into 6 equilateral triangles. Therefore:
- The area of the hexagon is [tex]\( 6 \)[/tex] times the area of one equilateral triangle.
So:
[tex]\[ \text{Area of hexagon} = 6 \times 4 \sqrt{3} \][/tex]
Simplifying this:
[tex]\[ \text{Area of hexagon} = 24 \sqrt{3} \text{ square feet} \][/tex]
### Step 2: Calculate the Volume of the Pyramid
Given:
- The height of the pyramid is [tex]\( 10 \)[/tex] feet.
- The base area of the hexagon calculated is [tex]\( 24 \sqrt{3} \)[/tex] square feet.
The formula for the volume of a pyramid is:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
So, substituting the values:
[tex]\[ V = \frac{1}{3} \times 24 \sqrt{3} \times 10 \][/tex]
Simplifying this:
[tex]\[ V = \frac{1}{3} \times 240 \sqrt{3} \][/tex]
[tex]\[ V = 80 \sqrt{3} \text{ cubic feet} \][/tex]
### Conclusion
The solutions are:
- The area of the regular hexagon base is [tex]\( \boxed{24 \sqrt{3}} \)[/tex] square feet.
- The volume of the pyramid is [tex]\( \boxed{80 \sqrt{3}} \)[/tex] cubic feet.
To relate this with the numerical results that were provided:
- The area [tex]\( 24 \sqrt{3} \)[/tex] square feet is approximately [tex]\( 41.57 \)[/tex] square feet.
- The volume [tex]\( 80 \sqrt{3} \)[/tex] cubic feet is approximately [tex]\( 138.56 \)[/tex] cubic feet.
### Step 1: Calculate the Area of the Regular Hexagon Base
Given:
- The base edge length of the hexagon is [tex]\( 4 \)[/tex] feet.
- The area of an equilateral triangle with sides [tex]\( 4 \)[/tex] feet is [tex]\( 4 \sqrt{3} \)[/tex] square feet.
A regular hexagon can be divided into 6 equilateral triangles. Therefore:
- The area of the hexagon is [tex]\( 6 \)[/tex] times the area of one equilateral triangle.
So:
[tex]\[ \text{Area of hexagon} = 6 \times 4 \sqrt{3} \][/tex]
Simplifying this:
[tex]\[ \text{Area of hexagon} = 24 \sqrt{3} \text{ square feet} \][/tex]
### Step 2: Calculate the Volume of the Pyramid
Given:
- The height of the pyramid is [tex]\( 10 \)[/tex] feet.
- The base area of the hexagon calculated is [tex]\( 24 \sqrt{3} \)[/tex] square feet.
The formula for the volume of a pyramid is:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
So, substituting the values:
[tex]\[ V = \frac{1}{3} \times 24 \sqrt{3} \times 10 \][/tex]
Simplifying this:
[tex]\[ V = \frac{1}{3} \times 240 \sqrt{3} \][/tex]
[tex]\[ V = 80 \sqrt{3} \text{ cubic feet} \][/tex]
### Conclusion
The solutions are:
- The area of the regular hexagon base is [tex]\( \boxed{24 \sqrt{3}} \)[/tex] square feet.
- The volume of the pyramid is [tex]\( \boxed{80 \sqrt{3}} \)[/tex] cubic feet.
To relate this with the numerical results that were provided:
- The area [tex]\( 24 \sqrt{3} \)[/tex] square feet is approximately [tex]\( 41.57 \)[/tex] square feet.
- The volume [tex]\( 80 \sqrt{3} \)[/tex] cubic feet is approximately [tex]\( 138.56 \)[/tex] cubic feet.
