Question

A grain silo can be modeled as a right cylinder topped with a hemisphere. Find the volume of the silo if it has a height of 69 m and a radius of 12 m. Round your answer to the nearest tenth if necessary

Answer 1

To find the volume of the silo, which consists of a right cylinder topped with a hemisphere, we first need to calculate the volumes of both shapes and then add them together.

### Volume of the Cylinder

The volume \( V_c \) of a cylinder is given by the formula:

\[
V_c = \pi r^2 h
\]

where:
- \( r \) is the radius of the cylinder,
- \( h \) is the height of the cylinder.

In this case, the radius \( r = 12 \) m and the height of the cylinder can be found by subtracting the height of the hemisphere from the total height of the silo.

The height of the hemisphere is equal to its radius:

\[
\text{Height of the hemisphere} = r = 12 \text{ m}
\]

Thus, the height of the cylinder \( h \) is:

\[
h = \text{Total height} - \text{Height of the hemisphere} = 69 \text{ m} - 12 \text{ m} = 57 \text{ m}
\]

Now we can calculate the volume of the cylinder:

\[
V_c = \pi (12^2)(57)
\]
\[
= \pi (144)(57)
\]
\[
= 8208\pi \text{ m}^3
\]

### Volume of the Hemisphere

The volume \( V_h \) of a hemisphere is given by the formula:

\[
V_h = \frac{2}{3} \pi r^3
\]

Using the radius \( r = 12 \) m:

\[
V_h = \frac{2}{3} \pi (12^3)
\]
\[
= \frac{2}{3} \pi (1728)
\]
\[
= \frac{3456}{3} \pi
\]
\[
= 1152\pi \text{ m}^3
\]

### Total Volume of the Silo

Now, we can find the total volume \( V \) of the silo by adding the volumes of the cylinder and the hemisphere:

\[
V = V_c + V_h
\]
\[
= 8208\pi + 1152\pi
\]
\[
= (8208 + 1152)\pi
\]
\[
= 9360\pi \text{ m}^3
\]

### Calculating Numeric Volume

To get an approximate value, we use \( \pi \approx 3.14159 \):

\[
V \approx 9360 \times 3.14159 \approx 29349.57384
\]

Rounding to the nearest tenth, we find:

\[
V \approx 29349.6 \text{ m}^3
\]

Thus, the volume of the silo is approximately:

\[
\boxed{29349.6 \text{ m}^3}
\]