In the measurement of volume of solid sphere using the formula `V=(4)/(3) pir^(3)` if the error committed in the measurement of the radius r is 2%, the percentage error in the volume measurement is

Options

A. 0.03
B. `1.5%`
C. `4.5%`
D. `6%`

Answer

To solve the problem of finding the percentage error in the volume measurement of a solid sphere given a 2% error in the radius measurement, we can follow these steps:

### Step 1: Understand the formula for volume
The volume \( V \) of a solid sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Identify the error in radius
We are given that the error in the measurement of the radius \( r \) is 2%. This means:
\[ \frac{\Delta r}{r} \times 100 = 2\% \]
where \( \Delta r \) is the absolute error in the radius.

### Step 3: Differentiate the volume formula
To find the relationship between the error in volume \( \Delta V \) and the error in radius \( \Delta r \), we differentiate the volume with respect to \( r \):
\[ \Delta V = \frac{dV}{dr} \Delta r \]
Calculating the derivative:
\[ \frac{dV}{dr} = 4 \pi r^2 \]
Thus,
\[ \Delta V = 4 \pi r^2 \Delta r \]

### Step 4: Express the percentage error in volume
The percentage error in volume \( \frac{\Delta V}{V} \) can be expressed as:
\[ \frac{\Delta V}{V} = \frac{4 \pi r^2 \Delta r}{\frac{4}{3} \pi r^3} \]
Simplifying this gives:
\[ \frac{\Delta V}{V} = \frac{3 \Delta r}{r} \]

### Step 5: Substitute the percentage error in radius
We know that:
\[ \frac{\Delta r}{r} = \frac{2}{100} = 0.02 \]
Substituting this into the equation for percentage error in volume:
\[ \frac{\Delta V}{V} = 3 \times 0.02 = 0.06 \]

### Step 6: Convert to percentage
To convert this to a percentage, we multiply by 100:
\[ \text{Percentage error in volume} = 0.06 \times 100 = 6\% \]

### Final Answer
The percentage error in the volume measurement is **6%**.

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