Two cylinddrical pots contain the same amount of water if their diameter are in the ratio `2:3` then what is the ratio of their heights ?

Answer

To solve the problem step by step, we will find the ratio of the heights of two cylindrical pots given that their diameters are in the ratio of 2:3.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two cylindrical pots (let's call them Pot P1 and Pot P2) that contain the same amount of water. The diameters of these pots are in the ratio 2:3.

2. **Let the Diameters be Represented**:
Let the diameter of Pot P1 be \(2x\) and the diameter of Pot P2 be \(3x\), where \(x\) is a common multiplier.

3. **Finding the Radii**:
The radius \(r\) of a cylinder is half of its diameter. Therefore:
- Radius of Pot P1, \(r_1 = \frac{2x}{2} = x\)
- Radius of Pot P2, \(r_2 = \frac{3x}{2} = \frac{3x}{2}\)

4. **Volume of the Cylinders**:
The formula for the volume \(V\) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \(r\) is the radius and \(h\) is the height.

- Volume of Pot P1:
\[ V_1 = \pi (r_1)^2 h_1 = \pi (x)^2 h_1 = \pi x^2 h_1 \]
- Volume of Pot P2:
\[ V_2 = \pi (r_2)^2 h_2 = \pi \left(\frac{3x}{2}\right)^2 h_2 = \pi \left(\frac{9x^2}{4}\right) h_2 = \frac{9\pi x^2}{4} h_2 \]

5. **Setting the Volumes Equal**:
Since both pots contain the same amount of water, we can set their volumes equal to each other:
\[ \pi x^2 h_1 = \frac{9\pi x^2}{4} h_2 \]

6. **Canceling Common Terms**:
We can cancel \(\pi\) and \(x^2\) from both sides (assuming \(x \neq 0\)):
\[ h_1 = \frac{9}{4} h_2 \]

7. **Finding the Ratio of Heights**:
To find the ratio of the heights \(h_1\) and \(h_2\):
\[ \frac{h_1}{h_2} = \frac{9}{4} \]

### Final Answer:
The ratio of the heights of the two pots is \(9:4\).

---