The density of a cube is found by measuring its mass and the length of its side of the maximum errors in the measurement of mass and length are 0.3% and 0.2% respectively, the maximum error in the measurement of density is

Options

A. `0.3%`
B. `0.5%`
C. `0.9%`
D. `1.1%`

Answer

To find the maximum error in the measurement of density based on the given errors in mass and length, we can follow these steps:

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

### Step 2: Identify the errors in measurement
From the problem, we know:
- The maximum error in the measurement of mass (\(m\)) is 0.3%.
- The maximum error in the measurement of length (\(l\)) is 0.2%.

### Step 3: Apply the formula for maximum error in density
To find the maximum percentage error in density, we use the formula for the propagation of errors:
\[ \text{Percentage error in } \rho = \left(1 \times \text{Percentage error in } m\right) + \left(3 \times \text{Percentage error in } l\right) \]
Here, the exponent of \(m\) is 1 and the exponent of \(l\) (since it is in the denominator as \(l^3\)) is 3.

### Step 4: Substitute the values into the formula
Substituting the given percentage errors:
\[ \text{Percentage error in } \rho = 1 \times 0.3\% + 3 \times 0.2\% \]

### Step 5: Calculate the total percentage error
Now, perform the calculations:
\[ \text{Percentage error in } \rho = 0.3\% + 0.6\% \]
\[ \text{Percentage error in } \rho = 0.9\% \]

### Conclusion
The maximum error in the measurement of density is:
\[ \text{Maximum error in density} = 0.9\% \]