Let's solve this problem step-by-step:
1. Understanding the Geometry:
- We have a container made up of two parts: a circular cylinder and a right circular cone.
- The height of the cylinder is 23 meters.
- The height of the cone is 4 meters.
- Both the cylinder and the cone share the same diameter of 6 meters.
2. Calculating the Radius:
- Diameter of the cylinder and cone = 6 meters.
- Radius ([tex]\( r \)[/tex]) is half of the diameter.
[tex]\[
r = \frac{\text{diameter}}{2} = \frac{6}{2} = 3 \text{ meters}
\][/tex]
3. Volume of the Cylinder:
- The formula for the volume of a cylinder is:
[tex]\[
V_{cylinder} = \pi r^2 h
\][/tex]
- Here, [tex]\( r = 3 \text{ meters} \)[/tex] and [tex]\( h = 23 \text{ meters} \)[/tex].
[tex]\[
V_{cylinder} = \pi (3)^2 (23) = \pi \cdot 9 \cdot 23 = 207 \pi \text{ cubic meters}
\][/tex]
- Converting to numerical value:
[tex]\[
V_{cylinder} \approx 650.3096792930871 \text{ cubic meters}
\][/tex]
4. Volume of the Cone:
- The formula for the volume of a cone is:
[tex]\[
V_{cone} = \frac{1}{3} \pi r^2 h
\][/tex]
- Here, [tex]\( r = 3 \text{ meters} \)[/tex] and [tex]\( h = 4 \text{ meters} \)[/tex].
[tex]\[
V_{cone} = \frac{1}{3} \pi (3)^2 (4) = \frac{1}{3} \pi \cdot 9 \cdot 4 = 12 \pi \text{ cubic meters}
\][/tex]
- Converting to numerical value:
[tex]\[
V_{cone} \approx 37.69911184307752 \text{ cubic meters}
\][/tex]
5. Total Volume of the Container:
- The total volume of the container is the sum of the volumes of the cylinder and the cone.
[tex]\[
V_{total} = V_{cylinder} + V_{cone}
\][/tex]
- Substituting the calculated volumes:
[tex]\[
V_{total} \approx 650.3096792930871 + 37.69911184307752 = 688.0087911361646 \text{ cubic meters}
\][/tex]
Therefore, the total volume of the container is approximately 688.0087911361646 cubic meters.
