The mass and sides of cube are given as (10 kg `+-0.1)` and `(0.1m+-0.01)` the fractional density is

Options

A. 0.31
B. 0.5
C. 0.62
D. 0.29

Answer

To find the fractional density of a cube given its mass and side length with uncertainties, we can follow these steps:

### Step 1: Understand the formula for density
The density (ρ) of a cube is defined as the mass (m) divided by the volume (V). For a cube with side length (a), the volume is given by:
\[ V = a^3 \]
Thus, the density can be expressed as:
\[ \rho = \frac{m}{a^3} \]

### Step 2: Identify the given values and their uncertainties
We are given:
- Mass \( m = 10 \, \text{kg} \) with an uncertainty \( \Delta m = 0.1 \, \text{kg} \)
- Side length \( a = 0.1 \, \text{m} \) with an uncertainty \( \Delta a = 0.01 \, \text{m} \)

### Step 3: Calculate the fractional density
The fractional density can be calculated using the formula for the relative error in density:
\[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3 \frac{\Delta a}{a} \]

### Step 4: Substitute the values into the formula
1. Calculate \( \frac{\Delta m}{m} \):
\[ \frac{\Delta m}{m} = \frac{0.1}{10} = 0.01 \]

2. Calculate \( \frac{\Delta a}{a} \):
\[ \frac{\Delta a}{a} = \frac{0.01}{0.1} = 0.1 \]

3. Now, substitute these values into the fractional density formula:
\[ \frac{\Delta \rho}{\rho} = 0.01 + 3 \times 0.1 = 0.01 + 0.3 = 0.31 \]

### Step 5: Conclusion
The fractional density is:
\[ \frac{\Delta \rho}{\rho} = 0.31 \]