The problem asks to find the approximate volume of a solid composed of a hemisphere on top of a cylinder. The diameter of the hemisphere and cylinder is 12 cm, and the height of the cylinder is 13 cm. The answer should be rounded to the nearest tenth.

1. Problem Description

The problem asks to find the approximate volume of a solid composed of a hemisphere on top of a cylinder. The diameter of the hemisphere and cylinder is 12 cm, and the height of the cylinder is 13 cm. The answer should be rounded to the nearest tenth.

2. Solution Steps

First, we need to find the radius of the hemisphere and cylinder. Since the diameter is 12 cm, the radius $r$ is half of that.

$r = \frac{12}{2} = 6$ cm

Next, we find the volume of the cylinder. The formula for the volume of a cylinder is:

$V_{cylinder} = \pi r^2 h$

Plugging in the values:

$V_{cylinder} = \pi (6)^2 (13) = \pi (36)(13) = 468\pi$

Now, we find the volume of the hemisphere. The formula for the volume of a sphere is $\frac{4}{3} \pi r^3$. Since a hemisphere is half of a sphere, the volume of the hemisphere is:

$V_{hemisphere} = \frac{1}{2} \cdot \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3$

Plugging in the values:

$V_{hemisphere} = \frac{2}{3} \pi (6)^3 = \frac{2}{3} \pi (216) = 144\pi$

The total volume is the sum of the volumes of the cylinder and the hemisphere:

$V_{total} = V_{cylinder} + V_{hemisphere} = 468\pi + 144\pi = 612\pi$

Using a calculator, $612\pi \approx 1922.6$

Rounding to the nearest tenth, we get 1922.

6.

3. Final Answer

1922.6