The density of a cube is measured by measuring its mass and length of its sides. If the maximum error in the measurement of mass and length are 4% and 3% respectively, the maximum error in the measurement of density will be

Options

A. `7%`
B. `9%`
C. `12%`
D. `13%`

Explanation

To solve the problem of finding the maximum error in the measurement of density given the maximum errors in mass and length, we can follow these steps:

### Step 1: Understand the formula for density
The formula for density (ρ) is given by:
\[ \rho = \frac{m}{V} \]
where \(m\) is the mass and \(V\) is the volume.

### Step 2: Determine the volume of the cube
For a cube, the volume \(V\) can be expressed in terms of the length of its sides \(L\):
\[ V = L^3 \]

### Step 3: Identify the errors in measurements
We are given:
- Maximum error in mass (\(m\)) = 4%
- Maximum error in length (\(L\)) = 3%

### Step 4: Calculate the maximum error in volume
Since the volume \(V\) depends on the length \(L\), we need to find the error in volume. The percentage error in volume can be calculated using the formula for the propagation of errors:
\[ \text{Percentage error in } V = 3 \times \text{Percentage error in } L \]
Substituting the given value:
\[ \text{Percentage error in } V = 3 \times 3\% = 9\% \]

### Step 5: Calculate the total maximum error in density
Now, we can calculate the maximum error in density using the formula for the propagation of errors:
\[ \text{Percentage error in } \rho = \text{Percentage error in } m + \text{Percentage error in } V \]
Substituting the known values:
\[ \text{Percentage error in } \rho = 4\% + 9\% = 13\% \]

### Step 6: State the final result
Thus, the maximum error in the measurement of density is:
\[ \text{Maximum error in density} = \pm 13\% \]

### Summary
The maximum error in the measurement of density is ±13%.

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