The error in measuring the side of a cube is `pm1%`. The error in the calculation of the volume of the cube will be about

Options

A. `pm 0.001 %`
B. `pm 1%`
C. `pm 6%`
D. `pm 3%`

Explanation

To solve the problem of finding the error in the calculation of the volume of a cube when the error in measuring the side of the cube is ±1%, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Volume Formula**:
The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
where \( a \) is the length of a side of the cube.

2. **Differentiate the Volume with Respect to Side Length**:
To find the relationship between the error in volume and the error in side length, we can take the logarithm of both sides:
\[ \log V = 3 \log a \]
Differentiating both sides gives:
\[ \frac{dV}{V} = 3 \frac{da}{a} \]

3. **Express the Errors**:
Let \( \delta V \) be the error in volume and \( \delta a \) be the error in measuring the side length. The relative error in volume can be expressed as:
\[ \frac{\delta V}{V} = 3 \frac{\delta a}{a} \]

4. **Convert to Percentage Error**:
To convert the relative error into percentage error, we multiply by 100:
\[ \frac{\delta V}{V} \times 100 = 3 \left( \frac{\delta a}{a} \times 100 \right) \]

5. **Substitute the Given Error**:
We know that the error in measuring the side length \( \frac{\delta a}{a} \times 100 \) is given as ±1%. Substituting this into our equation:
\[ \frac{\delta V}{V} \times 100 = 3 \times (\pm 1\%) \]

6. **Calculate the Error in Volume**:
Therefore, the percentage error in volume is:
\[ \frac{\delta V}{V} \times 100 = \pm 3\% \]

### Final Answer:
The error in the calculation of the volume of the cube will be about ±3%.

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