The error in the measurement of radius of a sphere is 0.4%. The percentage error in its volume is

Options

A. 0.16%
B. 0.12%
C. 0.24%
D. 1.2%

Answer

To solve the problem of finding the percentage error in the volume of a sphere given the percentage error in its radius, we can follow these steps:

### Step 1: Understand the given information
We are given that the error in the measurement of the radius of a sphere is 0.4%. This means that the relative error in the radius (r) can be expressed as:
\[ \frac{\Delta r}{r} \times 100 = 0.4 \]

### Step 2: Convert the percentage error to a decimal
To work with the relative error in calculations, we convert the percentage error into a decimal form:
\[ \text{Relative error in radius} = \frac{0.4}{100} = 0.004 \]

### Step 3: Write the formula for the volume of a sphere
The volume (V) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 4: Use the relationship between relative errors
According to the rules of propagation of errors, if \(x = a^n\), then the relative error in \(x\) is given by:
\[ \text{Relative error in } x = n \times \text{Relative error in } a \]
In our case, since volume is proportional to \(r^3\), we have:
\[ \text{Relative error in volume} = 3 \times \text{Relative error in radius} \]

### Step 5: Calculate the relative error in volume
Substituting the relative error in radius into the equation:
\[ \text{Relative error in volume} = 3 \times 0.004 = 0.012 \]

### Step 6: Convert the relative error in volume to percentage
To find the percentage error in volume, we multiply the relative error by 100:
\[ \text{Percentage error in volume} = 0.012 \times 100 = 1.2\% \]

### Conclusion
Thus, the percentage error in the volume of the sphere is **1.2%**.

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