Susan pours water from a full 4 L container into a number of water bottles for a camp hike.
Each water bottle is a cylinder with radius 4 cm and height 20 cm . How many bottles can be ﬁ lled
completely?
10 There are 80 liquorice cubes stacked in a cylindrical glass jar. The liquorice cubes have a side
length of 2 cm and the glass jar has a radius of 5 cm and a height of 12 cm . How much air space
remains in the jar of liquorice cubes? Give the answer correct to two decimal places.

Let's start by solving the two parts of the problem step by step.

### Part 1: Number of Water Bottles Filled

1. **Volume of Water in the Container:**
   - The container is 4 liters.
   - 1 liter = 1000 cm³
   - Therefore, 4 liters = \( 4 \times 1000 = 4000 \) cm³.

2. **Volume of One Water Bottle:**
   - Each water bottle is a cylinder with a radius \( r = 4 \) cm and height \( h = 20 \) cm.
   - The volume \( V \) of a cylinder is given by the formula \( V = \pi r^2 h \).
   - Substituting the given values:
     \[
     V = \pi \times (4 \, \text{cm})^2 \times 20 \, \text{cm}
     \]
     \[
     V = \pi \times 16 \, \text{cm}^2 \times 20 \, \text{cm}
     \]
     \[
     V = 320\pi \, \text{cm}^3
     \]

3. **Number of Bottles Filled:**
   - To find the number of bottles that can be completely filled, we divide the total volume of water by the volume of one bottle.
     \[
     \text{Number of bottles} = \frac{4000 \, \text{cm}^3}{320\pi \, \text{cm}^3}
     \]
   - Using \( \pi \approx 3.14159 \):
     \[
     \text{Number of bottles} = \frac{4000}{320 \times 3.14159}
     \]
     \[
     \text{Number of bottles} \approx \frac{4000}{1005.312}
     \]
     \[
     \text{Number of bottles} \approx 3.98
     \]
   - Since we can't fill a fraction of a bottle completely, the number of completely filled bottles is 3.

### Part 2: Air Space in the Jar of Liquorice Cubes

1. **Volume of the Glass Jar:**
   - The glass jar is a cylinder with a radius \( r = 5 \) cm and height \( h = 12 \) cm.
   - The volume \( V \) of the jar is:
     \[
     V = \pi r^2 h
     \]
     \[
     V = \pi \times (5 \, \text{cm})^2 \times 12 \, \text{cm}
     \]
     \[
     V = 300\pi \, \text{cm}^3
     \]
     \[
     V \approx 942.48 \, \text{cm}^3
     \]

2. **Volume of One Liquorice Cube:**
   - Each liquorice cube has a side length of 2 cm.
   - The volume \( V \) of one cube is:
     \[
     V = (2 \, \text{cm})^3 = 8 \, \text{cm}^3
     \]

3. **Total Volume of Liquorice Cubes:**
   - There are 80 liquorice cubes.
   - The total volume is:
     \[
     \text{Total volume of cubes} = 80 \times 8 \, \text{cm}^3 = 640 \, \text{cm}^3
     \]

4. **Air Space in the Jar:**
   - The air space is the difference between the volume of the jar and the total volume of the liquorice cubes.
     \[
     \text{Air space} = 942.48 \, \text{cm}^3 - 640 \, \text{cm}^3
     \]
     \[
     \text{Air space} = 302.48 \, \text{cm}^3
     \]

Thus, the air space remaining in the jar is \( 302.48 \) cm³.

### Summary of Results:
- Number of completely filled water bottles: 3
- Air space remaining in the jar: \( 302.48 \) cm³

Would you like more details or have any questions? 

Here are some related questions you might find interesting:

1. How would the number of bottles filled change if the bottle radius was doubled?
2. What is the effect of reducing the bottle height by half on the number of bottles filled?
3. How much water remains if 3 bottles are filled?
4. How many liquorice cubes can be added to the jar if the radius is increased by 2 cm?
5. What is the air space if the jar height is increased to 15 cm?
6. Calculate the number of water bottles that can be filled if the container holds 5 liters.
7. Determine the total surface area of all 80 liquorice cubes.
8. Find the volume of water each bottle would hold if the height was increased to 30 cm.

**Tip:** When working with volumes of irregular shapes, consider breaking them down into simpler geometric components for easier calculations.

Learn how to calculate the number of water bottles that can be filled and the remaining air space in a jar of liquorice cubes. This problem involves geometry and volume calculations of cylinders. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation