A hemispherical bowl of internal radius 15 cm contains a liquid. The liquid is to be filled into cylindrical shaped bottles of diameter 5 cm height 6 cm The number bottles required to empty the bowl is

Options

A. 30
B. 40
C. 50
D. 60

Answer

To solve the problem of how many cylindrical bottles are needed to empty a hemispherical bowl, we will follow these steps:

### Step 1: Calculate the volume of the hemispherical bowl
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 internal radius of the bowl is 15 cm, we can substitute this value into the formula:

\[ V = \frac{2}{3} \pi (15)^3 \]

Calculating \( (15)^3 \):

\[ 15^3 = 3375 \]

Now substituting back into the volume formula:

\[ V = \frac{2}{3} \pi \times 3375 \]

Calculating \( \frac{2}{3} \times 3375 \):

\[ \frac{2}{3} \times 3375 = 2250 \]

Thus, the volume of the hemispherical bowl is:

\[ V = 2250 \pi \text{ cm}^3 \]

### Step 2: 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 cylindrical bottle is 5 cm, so the radius \( r \) is:

\[ r = \frac{5}{2} = 2.5 \text{ cm} \]

The height \( h \) of the bottle is given as 6 cm. Now we can substitute these values into the volume formula:

\[ V = \pi (2.5)^2 \times 6 \]

Calculating \( (2.5)^2 \):

\[ (2.5)^2 = 6.25 \]

Now substituting back into the volume formula:

\[ V = \pi \times 6.25 \times 6 \]

Calculating \( 6.25 \times 6 \):

\[ 6.25 \times 6 = 37.5 \]

Thus, the volume of one cylindrical bottle is:

\[ V = 37.5 \pi \text{ cm}^3 \]

### Step 3: Calculate the number of bottles required
Let \( x \) be the number of bottles required to empty the bowl. The total volume of the liquid in the bowl is equal to the volume of \( x \) bottles:

\[ 2250 \pi = x \times 37.5 \pi \]

We can cancel \( \pi \) from both sides:

\[ 2250 = x \times 37.5 \]

Now, solving for \( x \):

\[ x = \frac{2250}{37.5} \]

Calculating \( \frac{2250}{37.5} \):

\[ x = 60 \]

### Conclusion
The number of bottles required to empty the bowl is **60**.

---