Question

A semi sphere sits on top of a cylinder. The height of the cylinder is 22 feet and the diameter of the cylinder is 25 feet. What is the total volume. Use 3.14 for pi. Round your answer to the nearest whole number.

Answer 1

To find the total volume of the semicylinder on top of the cylinder, we can calculate the volume of both the cylinder and the semicylinder separately, then add them together.

**Step 1: Calculate the 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,
- \( h \) is the height.

The diameter of the cylinder is 25 feet, so the radius \( r \) is:

\[
r = \frac{25}{2} = 12.5 \text{ feet}
\]
The height \( h \) of the cylinder is 22 feet.

Now, substituting these values into the volume formula:

\[
V_{\text{cylinder}} = 3.14 \times (12.5)^2 \times 22
\]

Calculating \( (12.5)^2 \):

\[
(12.5)^2 = 156.25
\]

Now substitute that back into the volume formula:

\[
V_{\text{cylinder}} = 3.14 \times 156.25 \times 22
\]

Calculating \( 3.14 \times 156.25 \):

\[
3.14 \times 156.25 \approx 490.625
\]

Now multiply by 22:

\[
V_{\text{cylinder}} \approx 490.625 \times 22 \approx 10792.75
\]

**Step 2: Calculate the volume of the semi sphere.**

The formula for the volume \( V \) of a sphere is given by:

\[
V = \frac{4}{3} \pi r^3
\]

Since we need the volume of a semicircle (half of a sphere), we will use:

\[
V_{\text{semisphere}} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3
\]

Using the same radius \( r = 12.5 \) feet, we find:

\[
V_{\text{semisphere}} = \frac{2}{3} \times 3.14 \times (12.5)^3
\]

Calculating \( (12.5)^3 \):

\[
(12.5)^3 = 12.5 \times 12.5 \times 12.5 = 156.25 \times 12.5 = 1953.125
\]

Now substitute back into the volume formula:

\[
V_{\text{semisphere}} = \frac{2}{3} \times 3.14 \times 1953.125
\]

Calculating \( 3.14 \times 1953.125 \):

\[
3.14 \times 1953.125 \approx 6138.70875
\]

Now multiply by \( \frac{2}{3} \):

\[
V_{\text{semisphere}} \approx \frac{2}{3} \times 6138.70875 \approx 4092.4725
\]

**Step 3: Calculate the total volume.**

Now, add the volume of the cylinder and the volume of the semisphere:

\[
V_{\text{total}} = V_{\text{cylinder}} + V_{\text{semisphere}} \approx 10792.75 + 4092.4725 \approx 14885.2225
\]

Rounding to the nearest whole number gives:

\[
\boxed{14885}
\]

Thus, the total volume is approximately 14,885 cubic feet.