To find the total volume of the solid consisting of a cylinder topped with a semi-sphere, we need to calculate the volume of each part separately and then sum them.
1. Volume of the Cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the base, and \( h \) is the height.
- Diameter of the cylinder = 25 feet, so the radius \( r \) is:
\[ r = \frac{25}{2} = 12.5 \text{ feet} \]
- Height \( h \) of the cylinder = 27 feet.
Now substituting the values into the volume formula:
\[ V_{cylinder} = \pi (12.5)^2 (27) \]
Calculating \( (12.5)^2 \):
\[ (12.5)^2 = 156.25 \]
Now, substituting that back into the volume formula:
\[ V_{cylinder} = \pi (156.25) (27) \]
Calculating \( (156.25)(27) \):
\[ 156.25 \times 27 = 4218.75 \]
Now, substituting this into the volume formula while using \( \pi \approx 3.14 \):
\[ V_{cylinder} = 3.14 \times 4218.75 \approx 13264.125 \]
2. Volume of the Semi-Sphere
The formula for the volume \( V \) of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
Since we need the volume of a semi-sphere, we take half of the volume of the sphere:
\[ V_{semi-sphere} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \]
The radius \( r \) of the semi-sphere is the same as that of the cylinder, which is 12.5 feet.
Now substituting that into the volume formula:
\[ V_{semi-sphere} = \frac{2}{3} \pi (12.5)^3 \]
Calculating \( (12.5)^3 \):
\[ (12.5)^3 = 12.5 \times 12.5 \times 12.5 = 1953.125 \]
Now substituting back into the volume formula:
\[ V_{semi-sphere} = \frac{2}{3} \pi (1953.125) \]
Calculating \( \frac{2}{3} \times 1953.125 \):
\[ \frac{2}{3} \times 1953.125 \approx 1302.08333 \]
Now substituting that back with \( \pi \approx 3.14 \):
\[ V_{semi-sphere} = 3.14 \times 1302.08333 \approx 4099.626 \]
3. Total Volume
Now we add the volumes of the cylinder and semi-sphere:
\[ V_{total} = V_{cylinder} + V_{semi-sphere} \approx 13264.125 + 4099.626 \approx 17363.751 \]
Rounding to the nearest whole number:
\[ V_{total} \approx 17364 \]
Final Answer
The total volume of the solid is approximately 17364 cubic feet.