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Bot GPT 4-o mini To solve problems involving the volume of a cube and a sphere, you'll need to understand the formulas for each shape and how to apply them in various scenarios.
### Volume of a Cube
The formula for the volume \( V \) of a cube is given by:
\[
V = s^3
\]
where \( s \) is the length of a side of the cube.
**Example Problem:**
- **Problem:** A cube has a side length of 4 cm. What is its volume?
- **Solution:**
\[
V = s^3 = 4^3 = 64 \text{ cm}^3
\]
### Volume of a Sphere
The formula for the volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
**Example Problem:**
- **Problem:** A sphere has a radius of 5 cm. What is its volume?
- **Solution:**
\[
V = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi \times 125 \approx 523.6 \text{ cm}^3
\]
### Applying the Formulas in Real-World Problems
1. **Finding Volume for Different Dimensions:**
- If a cube's side length changes, you can easily recalculate using the same formula.
- **Example:** If the side length changes to 10 cm:
\[
V = 10^3 = 1000 \text{ cm}^3
\]
2. **Comparing Volumes:**
- You can use these volume formulas to compare different shapes.
- **Example:** Compare the volume of a cube with a side of 4 cm and a sphere with a radius of 3 cm.
- Cube Volume:
\[
V_{\text{cube}} = 4^3 = 64 \text{ cm}^3
\]
- Sphere Volume:
\[
V_{\text{sphere}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \times 27 \approx 113.1 \text{ cm}^3
\]
- Conclusion: The volume of the cube is less than that of the sphere.
3. **Word Problems:**
- Real-world scenarios may involve finding the volume for various practical applications, such as how much water can fit in a container shaped like a cube or a sphere.
- **Example:** A tank in the shape of a cube can hold water. If the tank has a side of 2 m, how much water can it hold?
- Volume of water:
\[
V = 2^3 = 8 \text{ m}^3
\]
Using these formulas and procedures, you can tackle a wide range of problems related to cubes and spheres.