**A hemispherical bowl of internal diameter 54 cm contains a liquid. The liquid is to be filled in cylindrical bottles of radius 3 cm and height 9 cm. How many bottles are required to empty the bowl?**

221

343

81

243

None of these

We can solve this problem by comparing the volumes of the hemispherical bowl and the cylindrical bottles. To find how many bottles are required to empty the bowl, we need to:

1. **Calculate the volume of the hemispherical bowl**.
2. **Calculate the volume of one cylindrical bottle**.
3. **Divide the volume of the bowl by the volume of one bottle** to get the number of bottles required.

### Step 1: Volume of the Hemispherical Bowl
The formula for the volume of a hemisphere is:

\[
V_{\text{hemisphere}} = \frac{2}{3} \pi r^3
\]

The internal diameter of the bowl is given as 54 cm, so the radius \( r \) is:

\[
r = \frac{54}{2} = 27 \text{ cm}
\]

Thus, the volume of the bowl is:

\[
V_{\text{hemisphere}} = \frac{2}{3} \pi (27)^3
\]

\[
V_{\text{hemisphere}} = \frac{2}{3} \pi \times 19683
\]

\[
V_{\text{hemisphere}} = \frac{2}{3} \times 19683 \pi = 13122 \pi \text{ cubic cm}
\]

### Step 2: Volume of One Cylindrical Bottle
The formula for the volume of a cylinder is:

\[
V_{\text{cylinder}} = \pi r^2 h
\]

Where \( r = 3 \text{ cm} \) and \( h = 9 \text{ cm} \). Thus:

\[
V_{\text{cylinder}} = \pi (3)^2 \times 9
\]

\[
V_{\text{cylinder}} = \pi \times 9 \times 9 = 81 \pi \text{ cubic cm}
\]

### Step 3: Number of Bottles Required
The number of bottles is the ratio of the volume of the bowl to the volume of one bottle:

\[
\text{Number of bottles} = \frac{V_{\text{hemisphere}}}{V_{\text{cylinder}}} = \frac{13122 \pi}{81 \pi}
\]

Canceling out \( \pi \):

\[
\text{Number of bottles} = \frac{13122}{81} = 162
\]

Thus, **162 bottles** are required to empty the bowl.

### Final Answer: None of these (since 162 is not an option).

Would you like further details or clarifications?

Here are 5 related questions to consider:
1. What is the volume of a full sphere with a diameter of 54 cm?
2. How does the height of the liquid affect the number of bottles?
3. What if the diameter of the bottles was increased to 6 cm?
4. How would the answer change if the bowl was a full sphere?
5. Can we calculate the surface area of the hemispherical bowl?

**Tip**: Always ensure units are consistent when calculating volumes or areas to avoid errors.

A hemispherical bowl of internal diameter 54 cm contains a liquid. The liquid is to be filled in cylindrical bottles of radius 3 cm and height 9 cm. How many bottles are required to empty the bowl?

Calculate how many cylindrical bottles are required to empty a hemispherical bowl with an internal diameter of 54 cm. This problem covers key geometry concepts, including the volume formulas for hemispheres and cylinders. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation