The radius of a ball is `(5.2pm0.2)` cm. The percentage error in the volume of the ball is (approximately).

Options

A. 0.11
B. 0.04
C. 0.07
D. 0.09

Explanation

To find the percentage error in the volume of the ball given the radius and its uncertainty, we can follow these steps:

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

### Step 2: Identify the radius and its uncertainty
The radius of the ball is given as \( r = 5.2 \) cm with an uncertainty of \( \Delta r = 0.2 \) cm.

### Step 3: Determine the formula for percentage error in volume
The percentage error in volume can be calculated using the formula:
\[ \text{Percentage Error in } V = \frac{\Delta V}{V} \times 100 \]
where \( \Delta V \) is the uncertainty in volume.

### Step 4: Use the relationship between the radius and volume to find the uncertainty in volume
For small errors, the percentage error in volume can be approximated using the formula:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]
This is because the volume is proportional to the cube of the radius.

### Step 5: Substitute the values into the formula
Substituting the values we have:
\[ \Delta r = 0.2 \text{ cm}, \quad r = 5.2 \text{ cm} \]
We can calculate:
\[ \frac{\Delta V}{V} = 3 \frac{0.2}{5.2} \]

### Step 6: Calculate the percentage error
Now, we compute:
\[ 3 \frac{0.2}{5.2} = \frac{0.6}{5.2} \approx 0.11538 \]
To convert this to a percentage, we multiply by 100:
\[ \text{Percentage Error in } V \approx 0.11538 \times 100 \approx 11.538\% \]

### Step 7: Round the answer
Rounding to two decimal places, the percentage error in volume is approximately:
\[ \text{Percentage Error in } V \approx 11.54\% \]

### Final Answer
The percentage error in the volume of the ball is approximately \( 11.54\% \).

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