The mass of a cube of side 2cm is given as 100g. The side is accurate to the nearest millimetre and the mass accurate to the nearest gram. Calculate the maximum and minimum possible values for the density of the material (density = mass volume).

To solve this problem, we need to calculate the maximum and minimum possible values for the density of the cube's material, using the formula for density:

\[
\text{Density} = \frac{\text{Mass}}{\text{Volume}}
\]

### Given:
- Side of the cube \( s = 2 \) cm, accurate to the nearest millimeter.
- Mass of the cube \( m = 100 \) g, accurate to the nearest gram.

### Step 1: Determine the maximum and minimum possible values for the side length.
Since the side length is accurate to the nearest millimeter, the actual side length can vary by ±0.05 cm (0.05 cm is half of 1 mm).

- Maximum side length \( s_{\text{max}} = 2 \text{ cm} + 0.05 \text{ cm} = 2.05 \text{ cm} \)
- Minimum side length \( s_{\text{min}} = 2 \text{ cm} - 0.05 \text{ cm} = 1.95 \text{ cm} \)

### Step 2: Calculate the maximum and minimum possible values for the volume.
The volume \( V \) of a cube is given by:

\[
V = s^3
\]

- Maximum volume \( V_{\text{max}} = (2.05 \text{ cm})^3 = 2.05^3 = 8.615125 \text{ cm}^3 \)
- Minimum volume \( V_{\text{min}} = (1.95 \text{ cm})^3 = 1.95^3 = 7.418625 \text{ cm}^3 \)

### Step 3: Determine the maximum and minimum possible values for the mass.
Since the mass is accurate to the nearest gram, the actual mass can vary by ±0.5 grams.

- Maximum mass \( m_{\text{max}} = 100 \text{ g} + 0.5 \text{ g} = 100.5 \text{ g} \)
- Minimum mass \( m_{\text{min}} = 100 \text{ g} - 0.5 \text{ g} = 99.5 \text{ g} \)

### Step 4: Calculate the maximum and minimum possible values for the density.
Finally, use the formula for density with the maximum and minimum values for mass and volume:

1. **Maximum Density**:

\[
\text{Density}_{\text{max}} = \frac{m_{\text{max}}}{V_{\text{min}}} = \frac{100.5 \text{ g}}{7.418625 \text{ cm}^3} \approx 13.55 \text{ g/cm}^3
\]

2. **Minimum Density**:

\[
\text{Density}_{\text{min}} = \frac{m_{\text{min}}}{V_{\text{max}}} = \frac{99.5 \text{ g}}{8.615125 \text{ cm}^3} \approx 11.55 \text{ g/cm}^3
\]

### Final Answer:
- **Maximum Possible Density:** \( \approx 13.55 \text{ g/cm}^3 \)
- **Minimum Possible Density:** \( \approx 11.55 \text{ g/cm}^3 \)

Would you like more details on any of these steps?

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**Related Questions:**
1. How would the density change if the side length was measured in inches?
2. What is the relationship between volume and density for different geometric shapes?
3. How would rounding errors affect the density calculation for larger objects?
4. What are some practical applications of density in material science?
5. How would the calculation change if the mass had a different accuracy level?

**Tip:** Always check the accuracy of your measuring instruments to ensure the reliability of your calculated results!

Learn how to calculate the density of a cube with a given mass and side length accurate to the nearest millimeter and gram. Understand the concepts of density, volume, and accuracy in measurement. Suitable for high school students studying physics or mathematics.

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