The inner radius of a hemispherical utensil is 12 cm. This utensil is filled with the help of small cylindrical bottles. If the radius of the base of the bottle bee `3/2` cm and height by 4 cm. How many each bottles are required to fill it?

Options

A. 272
B. 64
C. 128
D. 136

Answer

To solve the problem of how many cylindrical bottles are required to fill a hemispherical utensil, we will follow these steps:

### Step 1: Calculate the volume of the hemispherical utensil.
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. Here, the inner radius of the hemispherical utensil is given as 12 cm.

Substituting \( r = 12 \) cm into the formula:

\[ V = \frac{2}{3} \pi (12)^3 \]

Calculating \( (12)^3 \):

\[ 12^3 = 1728 \]

Now substituting back into the volume formula:

\[ V = \frac{2}{3} \pi \times 1728 = \frac{3456}{3} \pi = 1152 \pi \, \text{cm}^3 \]

### 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 of the base and \( h \) is the height. The radius of the base of the bottle is \( \frac{3}{2} \) cm and the height is 4 cm.

Substituting these values into the formula:

\[ V = \pi \left(\frac{3}{2}\right)^2 \times 4 \]

Calculating \( \left(\frac{3}{2}\right)^2 \):

\[ \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

Now substituting back into the volume formula:

\[ V = \pi \times \frac{9}{4} \times 4 = \pi \times 9 = 9\pi \, \text{cm}^3 \]

### Step 3: Determine the number of bottles required.
Let \( n \) be the number of bottles required to fill the hemispherical utensil. We can set up the equation:

\[ n \times \text{Volume of one bottle} = \text{Volume of the hemisphere} \]

Substituting the volumes we calculated:

\[ n \times 9\pi = 1152\pi \]

Dividing both sides by \( \pi \):

\[ n \times 9 = 1152 \]

Now, solving for \( n \):

\[ n = \frac{1152}{9} = 128 \]

### Conclusion:
Therefore, the number of cylindrical bottles required to fill the hemispherical utensil is **128**.

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