The radius of a spherical ball is (10.4 `pm` 0.4) cm Select the correct alternative

Options

A. The percentage error in radius is 3.9%
B. The percentage error in radius is 0.4%
C. The percentage error in volume is 11. 5%
D. The absolute error in volume is 1.2 cm

Answer

To solve the problem, we need to find the percentage error in the radius and the volume of the spherical ball given its radius and the associated error.

### Step 1: Identify the given values
- Radius of the spherical ball, \( R = 10.4 \, \text{cm} \)
- Absolute error in radius, \( \Delta R = 0.4 \, \text{cm} \)

### Step 2: Calculate the percentage error in radius
The formula for percentage error is given by:

\[ \text{Percentage Error} = \left( \frac{\Delta R}{R} \right) \times 100 \]

Substituting the values:

\[ \text{Percentage Error in Radius} = \left( \frac{0.4}{10.4} \right) \times 100 \]

Calculating this:

\[ \text{Percentage Error in Radius} = \left( 0.03846 \right) \times 100 \approx 3.85\% \]

### Step 3: Calculate the volume of the sphere
The volume \( V \) of a sphere is given by the formula:

\[ V = \frac{4}{3} \pi R^3 \]

Substituting \( R = 10.4 \, \text{cm} \):

\[ V = \frac{4}{3} \pi (10.4)^3 \]

Calculating \( (10.4)^3 \):

\[ (10.4)^3 = 1121.664 \, \text{cm}^3 \]

Now substituting back to find \( V \):

\[ V \approx \frac{4}{3} \pi (1121.664) \approx 4690.56 \, \text{cm}^3 \]

### Step 4: Calculate the percentage error in volume
The volume is related to the radius cubed, so the percentage error in volume can be calculated using the formula:

\[ \frac{\Delta V}{V} = 3 \frac{\Delta R}{R} \]

Thus, the percentage error in volume is:

\[ \text{Percentage Error in Volume} = 3 \left( \frac{\Delta R}{R} \right) \times 100 \]

Substituting the values:

\[ \text{Percentage Error in Volume} = 3 \left( \frac{0.4}{10.4} \right) \times 100 \]

Calculating this:

\[ \text{Percentage Error in Volume} = 3 \times 3.85\% \approx 11.54\% \]

### Step 5: Check the options
Now, we can check the options provided in the question:
1. The % error in radius is 9.4% - **False**
2. The % error in radius is 0.4% - **False**
3. The % error in volume is 11.5% - **False**
4. The absolute error in volume is 1.2 cm - **False**

### Final Result
- Percentage error in radius: **3.85%**
- Percentage error in volume: **11.54%**