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 its radius and the uncertainty in that radius, 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 given values
From the problem, we have:
- Radius \( r = 5.2 \) cm
- Uncertainty in radius \( \Delta r = 0.2 \) cm

### Step 3: Calculate the relative error in radius
The relative error in the radius \( \frac{\Delta r}{r} \) can be calculated as:
\[ \frac{\Delta r}{r} = \frac{0.2}{5.2} \]

### Step 4: Calculate the relative error in volume
The volume of a sphere depends on the radius cubed. Therefore, the relative error in volume \( \frac{\Delta V}{V} \) is related to the relative error in radius by the formula:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{\Delta r}{r} \]

### Step 5: Substitute the values
Now substituting the values we have:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{0.2}{5.2} \]

### Step 6: Calculate the relative error in volume
Calculating this gives:
\[ \frac{\Delta V}{V} = 3 \cdot \frac{0.2}{5.2} = \frac{0.6}{5.2} \approx 0.1154 \]

### Step 7: Convert to percentage
To find the percentage error, we multiply the relative error by 100:
\[ \text{Percentage Error} = 0.1154 \times 100 \approx 11.54\% \]

### Step 8: Round to the nearest whole number
Since we are looking for an approximate percentage error, we can round this to:
\[ \text{Percentage Error} \approx 12\% \]

### Final Answer
The approximate percentage error in the volume of the ball is **12%**.

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