The radius of a sphere is measured as `(10 +- 0.02%)` cm. The error in the measurement of its volume is

Options

A. 25.1 cc
B. 25.12 cc
C. 2.51 cc
D. 251.2 cc

Answer

To find the error in the measurement of the volume of a sphere given the radius and its percentage error, we can follow these steps:

### Step 1: Understand the given data
The radius of the sphere is given as \( r = 10 \, \text{cm} \) with an error of \( \Delta r = 0.02\% \).

### Step 2: Calculate the absolute error in the radius
To find the absolute error in the radius, we can calculate:
\[ \Delta r = \frac{0.02}{100} \times r = \frac{0.02}{100} \times 10 = 0.002 \, \text{cm} \]

### 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: Calculate the volume using the radius
Substituting \( r = 10 \, \text{cm} \) into the volume formula:
\[ V = \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi \times 1000 = \frac{4000}{3} \pi \, \text{cm}^3 \]

### Step 5: Determine the relative error in volume
The relative error in volume \( \frac{\Delta V}{V} \) can be calculated using the formula:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{\Delta r}{r} \]
Substituting the values:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{0.002}{10} = 3 \cdot 0.0002 = 0.0006 \]

### Step 6: Calculate the absolute error in volume
Now, we can calculate the absolute error in volume \( \Delta V \):
\[ \Delta V = V \cdot \frac{\Delta V}{V} = V \cdot 0.0006 \]
Substituting the value of \( V \):
\[ \Delta V = \left(\frac{4000}{3} \pi\right) \cdot 0.0006 \]
Calculating \( V \):
\[ V \approx 4188.79 \, \text{cm}^3 \quad (\text{using } \pi \approx 3.14) \]
Now substituting:
\[ \Delta V \approx 4188.79 \cdot 0.0006 \approx 2.51 \, \text{cm}^3 \]

### Conclusion
Thus, the error in the measurement of the volume of the sphere is approximately \( 2.51 \, \text{cm}^3 \).