The radius of a sphere is `(5.3 +- 0.1)`cm` The perecentage error in its volume is

Options

A. `(0.1)/(5.3) xx 100`
B. `3 xx(0.1)/(5.3) xx 100`
C. `(0.1 xx 100)/(3.53)`
D. `3 +(0.1)/(5.3) xx 100`

Explanation

To find the percentage error in the volume of a sphere 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 given values
From the question, we have:
- Radius \( r = 5.3 \, \text{cm} \)
- Uncertainty in radius \( \Delta r = 0.1 \, \text{cm} \)

### Step 3: Calculate the volume of the sphere
First, we calculate the volume using the radius:
\[ V = \frac{4}{3} \pi (5.3)^3 \]
Calculating \( (5.3)^3 \):
\[ (5.3)^3 = 148.877 \]
Now substituting back into the volume formula:
\[ V = \frac{4}{3} \pi (148.877) \approx \frac{4}{3} \times 3.14 \times 148.877 \approx 623.24 \, \text{cm}^3 \]

### Step 4: Calculate the error in volume
To find the percentage error in volume, we use the formula for the propagation of uncertainty:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]
where \( \Delta V \) is the uncertainty in volume, and \( \Delta r \) is the uncertainty in radius.

### Step 5: Substitute the values into the error formula
Substituting the known values:
\[ \frac{\Delta V}{V} = 3 \frac{0.1}{5.3} \]
Calculating this:
\[ \frac{\Delta V}{V} = 3 \times \frac{0.1}{5.3} \approx 3 \times 0.01887 \approx 0.0566 \]

### Step 6: Calculate the percentage error
Now, to find the percentage error:
\[ \text{Percentage error} = \left( \frac{\Delta V}{V} \right) \times 100 \approx 0.0566 \times 100 \approx 5.66\% \]

### Final Answer
The percentage error in the volume of the sphere is approximately \( 5.66\% \).

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