A cylindrical vessel of height 24 cm and diamater 40 cm is full of water. Find the exact number of small cylindrical bottles, each of height 10 cm and diameter 8 cm, which can be filled with this water.

Answer

To find the exact number of small cylindrical bottles that can be filled with the water from a larger cylindrical vessel, we will follow these steps:

### Step 1: Calculate the volume of the larger cylindrical vessel.
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.

For the larger vessel:
- Diameter = 40 cm, so the radius \( r = \frac{40}{2} = 20 \) cm.
- Height \( h = 24 \) cm.

Substituting these values into the volume formula:
\[ V_{\text{large}} = \pi (20)^2 (24) = \pi (400)(24) = 9600\pi \text{ cm}^3 \]

### Step 2: Calculate the volume of one small cylindrical bottle.
Using the same volume formula for the small bottle:
- Diameter = 8 cm, so the radius \( r = \frac{8}{2} = 4 \) cm.
- Height \( h = 10 \) cm.

Substituting these values into the volume formula:
\[ V_{\text{small}} = \pi (4)^2 (10) = \pi (16)(10) = 160\pi \text{ cm}^3 \]

### Step 3: Determine the number of small bottles that can be filled.
Let \( n \) be the number of small bottles that can be filled with the water from the larger vessel. The total volume of the small bottles must equal the volume of the larger vessel:
\[ n \cdot V_{\text{small}} = V_{\text{large}} \]
Substituting the volumes we calculated:
\[ n \cdot 160\pi = 9600\pi \]

### Step 4: Solve for \( n \).
We can cancel \( \pi \) from both sides:
\[ n \cdot 160 = 9600 \]
Now, divide both sides by 160:
\[ n = \frac{9600}{160} = 60 \]

### Conclusion:
The exact number of small cylindrical bottles that can be filled with the water from the larger cylindrical vessel is **60**.

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