The relative error in the measurement of the side of a cube is 0.027 The relative error in the measurement of its volume is

Options

A. 0.027
B. 0.054
C. 0.081
D. 0.046

Answer

To find the relative error in the measurement of the volume of a cube given the relative error in the measurement of its side, we can follow these steps:

### Step 1: Understand the relationship between side and volume
The volume \( V \) of a cube with side length \( A \) is given by the formula:
\[ V = A^3 \]

### Step 2: Define relative error
The relative error in a measurement is defined as:
\[ \text{Relative Error} = \frac{\Delta A}{A} \]
where \( \Delta A \) is the absolute error in the measurement of \( A \).

### Step 3: Relate the relative error of volume to the side
We need to find the relative error in the volume \( V \). Using the formula for volume, we can express the relative error in volume in terms of the relative error in the side:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{\Delta A}{A} \]
This relationship comes from differentiating the volume with respect to the side length.

### Step 4: Substitute the known relative error
We are given that the relative error in the measurement of the side of the cube is:
\[ \frac{\Delta A}{A} = 0.027 \]
Now, substituting this value into the equation for the relative error in volume:
\[ \frac{\Delta V}{V} = 3 \cdot 0.027 \]

### Step 5: Calculate the relative error in volume
Now, we perform the multiplication:
\[ \frac{\Delta V}{V} = 3 \cdot 0.027 = 0.081 \]

### Conclusion
Thus, the relative error in the measurement of the volume of the cube is:
\[ \frac{\Delta V}{V} = 0.081 \]

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