If the percentage error in the edge of a cube is 1, then error in its volume, is

Options

A. 0.01
B. `2%`
C. `3%`
D. none of these

Explanation

To solve the problem of finding the error in the volume of a cube when the percentage error in the edge of the cube is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We are given that the percentage error in the edge (let's denote it as \( a \)) of the cube is \( 1\% \).

2. **Express the Percentage Error**:
- The percentage error in the edge can be expressed mathematically as:
\[ \text{Percentage Error} = \frac{\Delta a}{a} \times 100 \]
- Here, \( \Delta a \) is the absolute error in the edge length.

3. **Set Up the Equation**:
- Given that the percentage error is \( 1\% \), we can write:
\[ \frac{\Delta a}{a} \times 100 = 1 \]
- This simplifies to:
\[ \frac{\Delta a}{a} = \frac{1}{100} \]

4. **Volume of the Cube**:
- The volume \( V \) of the cube is given by:
\[ V = a^3 \]

5. **Differentiate the Volume**:
- To find the error in volume, we differentiate \( V \) with respect to \( a \):
\[ dV = 3a^2 \, da \]
- Here, \( dV \) is the change in volume and \( da \) is the change in edge length.

6. **Relate the Changes**:
- We can express the relative change in volume as:
\[ \frac{dV}{V} = \frac{3a^2 \, da}{a^3} \]
- Simplifying this gives:
\[ \frac{dV}{V} = \frac{3 \, da}{a} \]

7. **Calculate the Percentage Error in Volume**:
- The percentage error in volume is given by:
\[ \text{Percentage Error in Volume} = \frac{dV}{V} \times 100 \]
- Substituting our expression for \(\frac{dV}{V}\):
\[ \text{Percentage Error in Volume} = \left(3 \frac{da}{a}\right) \times 100 \]

8. **Substitute the Known Value**:
- We know from step 3 that \(\frac{da}{a} = \frac{1}{100}\):
\[ \text{Percentage Error in Volume} = 3 \times \left(\frac{1}{100}\right) \times 100 \]
- This simplifies to:
\[ \text{Percentage Error in Volume} = 3\% \]

### Final Result:
The error in the volume of the cube is \( 3\% \).

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