The lengths of sides of a cuboid are a, 2a and 3a. If the relative percentage error in the measurement of a is `1%`, then what is the relative percentage error in the measurement of the volume of the cuboid.

Answer

To solve the problem of finding the relative percentage error in the measurement of the volume of a cuboid with sides \(a\), \(2a\), and \(3a\), given that the relative percentage error in the measurement of \(a\) is \(1\%\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Volume Formula**:
The volume \(V\) of a cuboid is given by the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
For our cuboid, the sides are \(a\), \(2a\), and \(3a\). Thus, the volume can be calculated as:
\[ V = a \times 2a \times 3a = 6a^3 \]

2. **Determine the Differential of Volume**:
To find the relative error in volume, we need to differentiate the volume with respect to \(a\). Using the formula for volume:
\[ V = 6a^3 \]
The differential \(dV\) can be calculated as:
\[ dV = \frac{dV}{da} \cdot da = 18a^2 \cdot da \]

3. **Calculate the Relative Error**:
The relative error in volume is given by the formula:
\[ \frac{dV}{V} = \frac{18a^2 \cdot da}{6a^3} \]
Simplifying this expression:
\[ \frac{dV}{V} = \frac{18}{6} \cdot \frac{da}{a} = 3 \cdot \frac{da}{a} \]

4. **Substituting the Relative Percentage Error of \(a\)**:
We know that the relative percentage error in \(a\) is \(1\%\), which can be expressed as:
\[ \frac{da}{a} = \frac{1}{100} \]
Substituting this into the equation gives:
\[ \frac{dV}{V} = 3 \cdot \frac{1}{100} = \frac{3}{100} \]

5. **Convert to Percentage**:
To find the relative percentage error in volume, we multiply by \(100\):
\[ \text{Percentage error in volume} = \frac{3}{100} \times 100 = 3\% \]

### Final Answer:
The relative percentage error in the measurement of the volume of the cuboid is \(3\%\).