To find the volume of the concrete used to make the tower, we need to follow these steps:
1. Calculate the Apothem:
- The apothem (a) is the distance from the center of the regular octagon to the midpoint of one of its sides.
- The apothem of a regular octagon with a side length [tex]\( s \)[/tex] can be determined using the formula:
[tex]\[
a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)}
\][/tex]
- Given that the side length ([tex]\( s \)[/tex]) is 30 m, the apothem [tex]\( a \)[/tex] is:
[tex]\[
a = \frac{30}{2 \tan\left(\frac{\pi}{8}\right)} = 36.213 \text{ meters (approx)}
\][/tex]
2. Calculate the Area of One Triangle:
- The regular octagon can be divided into 8 identical isosceles triangles.
- The area of one triangle is given by:
[tex]\[
\text{Area of one triangle} = \frac{1}{2} \times \text{side length} \times \text{apothem}
\][/tex]
- Therefore, the area of one triangle is:
[tex]\[
\text{Area of one triangle} = \frac{1}{2} \times 30 \times 36.213 = 543.198 \text{ square meters (approx)}
\][/tex]
3. Calculate the Total Area of the Octagon:
- The total area of the octagon is the sum of the areas of 8 triangles:
[tex]\[
\text{Area of Octagon} = 8 \times \text{Area of one triangle}
\][/tex]
- Thus, the total area of the octagon is:
[tex]\[
\text{Area of Octagon} = 8 \times 543.198 = 4345.584 \text{ square meters (approx)}
\][/tex]
4. Calculate the Volume of the Octagonal Prism:
- The volume of a prism is given by the product of the base area and the height:
[tex]\[
\text{Volume} = \text{Base Area} \times \text{Height}
\][/tex]
- Given that the height ([tex]\( h \)[/tex]) is 24 m:
[tex]\[
\text{Volume} = 4345.584 \times 24 = 104294.026 \text{ cubic meters (approx)}
\][/tex]
In conclusion, the volume of the concrete used to make the octagonal tower is approximately 104294.026 cubic meters.
