A solid consisting of a right circular cone, standing on a hemisphere, is placed upright, in a right circular cylinder, full of water, and touches the bottom. Find the volume of water left in the cylinder, having given that the radius of the cylinder is 3 cm and its height is 6 cm, the radius of the hemisphere is 2 cm and the height of the cone is 4 cm. Give your answer correct to the nearest centimetre
( Take ` pi = 3 (1)/(7)) `

Answer

To solve the problem step by step, we will calculate the volume of the cylinder, the volume of the solid (which consists of a cone and a hemisphere), and then find the volume of water left in the cylinder.

### Step 1: Calculate the volume of the cylinder
The formula for the volume of a cylinder is given by:

\[ V_{\text{cylinder}} = \pi r^2 h \]

Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height of the cylinder

Given:
- Radius of the cylinder \( r = 3 \) cm
- Height of the cylinder \( h = 6 \) cm

Substituting these values into the formula:

\[ V_{\text{cylinder}} = \pi (3)^2 (6) = \pi (9)(6) = 54\pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the cone
The formula for the volume of a cone is:

\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

Where:
- \( r \) is the radius of the cone
- \( h \) is the height of the cone

Given:
- Radius of the cone \( r = 2 \) cm
- Height of the cone \( h = 4 \) cm

Substituting these values into the formula:

\[ V_{\text{cone}} = \frac{1}{3} \pi (2)^2 (4) = \frac{1}{3} \pi (4)(4) = \frac{16}{3}\pi \text{ cm}^3 \]

### Step 3: Calculate the volume of the hemisphere
The formula for the volume of a hemisphere is:

\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]

Where:
- \( r \) is the radius of the hemisphere

Given:
- Radius of the hemisphere \( r = 2 \) cm

Substituting this value into the formula:

\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (2)^3 = \frac{2}{3} \pi (8) = \frac{16}{3}\pi \text{ cm}^3 \]

### Step 4: Calculate the total volume of the solid
The total volume of the solid (cone + hemisphere) is:

\[ V_{\text{solid}} = V_{\text{cone}} + V_{\text{hemisphere}} = \frac{16}{3}\pi + \frac{16}{3}\pi = \frac{32}{3}\pi \text{ cm}^3 \]

### Step 5: Calculate the volume of water left in the cylinder
The volume of water left in the cylinder is calculated by subtracting the volume of the solid from the volume of the cylinder:

\[ V_{\text{water}} = V_{\text{cylinder}} - V_{\text{solid}} = 54\pi - \frac{32}{3}\pi \]

To perform this subtraction, we need a common denominator:

\[ 54\pi = \frac{162}{3}\pi \]

Now, substituting back into the equation:

\[ V_{\text{water}} = \frac{162}{3}\pi - \frac{32}{3}\pi = \frac{130}{3}\pi \text{ cm}^3 \]

### Step 6: Substitute \(\pi\) and calculate the final volume
Using \(\pi = \frac{22}{7}\):

\[ V_{\text{water}} = \frac{130}{3} \times \frac{22}{7} \]

Calculating this:

\[ V_{\text{water}} = \frac{130 \times 22}{3 \times 7} = \frac{2860}{21} \approx 136.19 \text{ cm}^3 \]

### Final Answer
Rounding to the nearest centimeter, the volume of water left in the cylinder is approximately:

\[ \text{Volume of water left} \approx 136 \text{ cm}^3 \]