To find the total volume of the semi-sphere sitting on top of the cylinder, we need to calculate the volumes of both the cylinder and the semi-sphere and then add them together.
Step 1: Calculate the volume of the cylinder.
The formula for the volume \( V \) of a cylinder is:
\[ V = \pi r^2 h \]
Where:
- \( r \) is the radius of the cylinder,
- \( h \) is the height of the cylinder.
Given:
- The diameter of the cylinder is 38 feet, so the radius \( r = \frac{38}{2} = 19 \) feet.
- The height \( h = 35 \) feet.
Now we can plug in the values:
\[ V_{\text{cylinder}} = 3.14 \times (19)^2 \times 35 \]
Calculating \( (19)^2 \):
\[ (19)^2 = 361 \]
Now substituting back into the volume formula:
\[ V_{\text{cylinder}} = 3.14 \times 361 \times 35 \]
Calculating \( 361 \times 35 \):
\[ 361 \times 35 = 12635 \]
Now, calculating the volume of the cylinder:
\[ V_{\text{cylinder}} = 3.14 \times 12635 \approx 39613.9 \]
Step 2: Calculate the 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 sphere's volume:
\[ V_{\text{semi-sphere}} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \]
Using the same radius \( r = 19 \) feet:
\[ V_{\text{semi-sphere}} = \frac{2}{3} \times 3.14 \times (19)^3 \]
Calculating \( (19)^3 \):
\[ (19)^3 = 6859 \]
Now substituting back into the volume formula:
\[ V_{\text{semi-sphere}} = \frac{2}{3} \times 3.14 \times 6859 \]
Calculating \( \frac{2}{3} \times 3.14 \):
\[ \frac{2}{3} \times 3.14 \approx 2.09333 \]
Now calculating the volume of the semi-sphere:
\[ V_{\text{semi-sphere}} = 2.09333 \times 6859 \approx 14312.4 \]
Step 3: Calculate the total volume.
Now we add the volume of the cylinder to the volume of the semi-sphere:
\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{semi-sphere}} \approx 39613.9 + 14312.4 \]
Calculating the total volume:
\[ V_{\text{total}} \approx 53926.3 \]
Rounding to the nearest whole number gives:
\[ V_{\text{total}} \approx 53926 \]
Thus, the total volume of the silo is 53926 cubic feet.