A hemispherical bowl of internal diameter 30 cm contains some liquid. This liquid is to be filled into cylindrical - shaped bottles each of diameter 5 cm and height 6 cm. Find the number of bottles necessary to empty the bowl.

Answer

To solve the problem, we need to calculate the volume of the hemispherical bowl and the volume of one cylindrical bottle. Then, we will determine how many bottles are needed to hold the liquid from the bowl.

### Step 1: Calculate the volume of the hemispherical bowl.

The formula for the volume of a hemisphere is given by:

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

Given that the internal diameter of the bowl is 30 cm, we can find the radius:

\[ r_1 = \frac{30}{2} = 15 \text{ cm} \]

Now, substituting the radius into the volume formula:

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

Calculating \(15^3\):

\[ 15^3 = 3375 \]

Thus,

\[ V_{hemisphere} = \frac{2}{3} \pi (3375) = \frac{6750}{3} \pi = 2250 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of one cylindrical bottle.

The formula for the volume of a cylinder is given by:

\[ V = \pi r^2 h \]

Given that the diameter of the bottle is 5 cm, we can find the radius:

\[ r_2 = \frac{5}{2} = 2.5 \text{ cm} \]

The height of the bottle is given as 6 cm. Now substituting the values into the volume formula:

\[ V_{cylinder} = \pi (2.5)^2 (6) \]

Calculating \( (2.5)^2 \):

\[ (2.5)^2 = 6.25 \]

Thus,

\[ V_{cylinder} = \pi (6.25) (6) = 37.5 \pi \text{ cm}^3 \]

### Step 3: Calculate the number of bottles needed.

To find the number of bottles needed to empty the bowl, we divide the volume of the hemisphere by the volume of one cylindrical bottle:

\[ \text{Number of bottles} = \frac{V_{hemisphere}}{V_{cylinder}} = \frac{2250 \pi}{37.5 \pi} \]

The \(\pi\) cancels out:

\[ \text{Number of bottles} = \frac{2250}{37.5} \]

Calculating this gives:

\[ \text{Number of bottles} = 60 \]

### Conclusion

Thus, the number of bottles necessary to empty the bowl is **60**.

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