A hemispherical bowl of internal radius 9 cm is full of liquid. This liquid is to be filled into conical shaped small containers each of diameter 3 cm and height 4 cm. How many containers are necessary to empty the bowl?

Answer

To solve the problem step by step, we need to find the volume of the hemispherical bowl and the volume of one conical container, and then determine how many conical containers are needed to hold the liquid from the bowl.

### 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 radius \( r = 9 \) cm, we can substitute this value into the formula:

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

Calculating \( 9^3 \):

\[ 9^3 = 729 \]

Now substituting back into the volume formula:

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

### Step 2: Calculate the volume of one conical container.

The formula for the volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

Where \( r \) is the radius and \( h \) is the height. The diameter of the conical container is given as 3 cm, so the radius \( r \) is:

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

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

\[ V = \frac{1}{3} \pi (1.5)^2 (4) \]

Calculating \( (1.5)^2 \):

\[ (1.5)^2 = 2.25 \]

Now substituting back into the volume formula:

\[ V = \frac{1}{3} \pi (2.25)(4) = \frac{1}{3} \pi (9) = 3 \pi \text{ cm}^3 \]

### Step 3: Determine the number of conical containers needed.

To find the number of conical containers required to empty the bowl, we divide the volume of the hemispherical bowl by the volume of one conical container:

\[ \text{Number of containers} = \frac{\text{Volume of hemispherical bowl}}{\text{Volume of conical container}} = \frac{486 \pi}{3 \pi} \]

The \( \pi \) cancels out:

\[ \text{Number of containers} = \frac{486}{3} = 162 \]

### Final Answer:

Therefore, the number of conical containers necessary to empty the bowl is **162**.

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