The inner radius of a hemispherical cup is 9 cm and it is completely full of water .This water is filled in cylindrical bottles of diameter 3cm and height 4 cm Find the number of bottles required.

Answer

To solve the problem, we need to find out how many cylindrical bottles can be filled with the water from a hemispherical cup. Here’s a step-by-step solution:

### Step 1: Calculate the volume of the hemispherical cup.
The formula for the volume \( V \) of a hemisphere is given by:

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

where \( r \) is the radius of the hemisphere. Given that the inner radius of the hemispherical cup is 9 cm, we can substitute this value into the formula:

\[ V = \frac{2}{3} \pi (9)^3 \]

### Step 2: Calculate \( 9^3 \).
Calculating \( 9^3 \):

\[ 9^3 = 729 \]

### Step 3: Substitute \( 729 \) back into the volume formula.
Now substituting \( 729 \) back into the volume formula:

\[ V = \frac{2}{3} \pi (729) = \frac{1458}{3} \pi = 486 \pi \, \text{cm}^3 \]

### Step 4: Calculate the volume of one cylindrical bottle.
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. The diameter of the bottle is given as 3 cm, so the radius \( r \) is:

\[ r = \frac{3}{2} = 1.5 \, \text{cm} \]

The height \( h \) of the bottle is given as 4 cm. Now, substituting these values into the volume formula:

\[ V = \pi (1.5)^2 (4) \]

### Step 5: Calculate \( (1.5)^2 \).
Calculating \( (1.5)^2 \):

\[ (1.5)^2 = 2.25 \]

### Step 6: Substitute \( 2.25 \) back into the volume formula for the cylinder.
Now substituting \( 2.25 \) back into the volume formula:

\[ V = \pi (2.25)(4) = 9 \pi \, \text{cm}^3 \]

### Step 7: Find the number of bottles required.
Let \( N \) be the number of bottles required. The total volume of water from the hemispherical cup is equal to the volume of \( N \) cylindrical bottles:

\[ 486 \pi = N \times 9 \pi \]

### Step 8: Cancel \( \pi \) from both sides.
Dividing both sides by \( \pi \):

\[ 486 = N \times 9 \]

### Step 9: Solve for \( N \).
Now, solving for \( N \):

\[ N = \frac{486}{9} = 54 \]

### Conclusion:
The number of bottles required is \( \boxed{54} \).

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