The error in the measurement of the radius of a sphere is 1%. The error in the measurement of volume is

Options

A. 0.01
B. 0.03
C. 0.05
D. 0.08

Explanation

To find the error in the measurement of the volume of a sphere when the error in the measurement of the radius is given, we can follow these steps:

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

### Step 2: Determine the relative error in volume
The relative error in volume can be calculated using the formula for the error propagation in functions of multiple variables. For the volume of a sphere, the relationship between the volume and the radius gives us:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]
where \( \Delta V \) is the change in volume, \( V \) is the original volume, \( \Delta r \) is the change in radius, and \( r \) is the original radius.

### Step 3: Substitute the given error in radius
We are given that the error in the measurement of the radius is 1%. This means:
\[ \frac{\Delta r}{r} = 0.01 \]

### Step 4: Calculate the error in volume
Substituting this value into the error propagation formula:
\[ \frac{\Delta V}{V} = 3 \times 0.01 = 0.03 \]

### Step 5: Convert relative error to percentage
To express the error in volume as a percentage, we multiply by 100:
\[ \text{Error in volume} = 0.03 \times 100 = 3\% \]

### Conclusion
The error in the measurement of the volume of the sphere is 3%.

### Final Answer
The error in the measurement of volume is **0.03** (or 3%).

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