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

Options

A. `0.1/5.3xx100`
B. `3xx0.1/5.3xx100`
C. `3/2xx0.1/5.3xx100`
D. `6xx0.1/0.3xx100`

Answer

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

### Step 2: Identify the given values
From the problem, we have:
- Radius \( r = 5.3 \, \text{cm} \)
- Uncertainty in radius \( dr = 0.1 \, \text{cm} \)

### Step 3: Calculate the relative error in radius
The relative error in the radius can be calculated using the formula:
\[ \frac{dr}{r} = \frac{0.1}{5.3} \]

### Step 4: Calculate the percentage error in radius
To convert the relative error to a percentage, multiply by 100:
\[ \text{Percentage error in radius} = \left( \frac{dr}{r} \right) \times 100 = \left( \frac{0.1}{5.3} \right) \times 100 \]

### Step 5: Relate the percentage error in volume to the percentage error in radius
The volume of a sphere is proportional to the cube of the radius. Therefore, the percentage error in volume \( \left( \frac{dV}{V} \right) \) is three times the percentage error in radius:
\[ \text{Percentage error in volume} = 3 \times \text{Percentage error in radius} \]

### Step 6: Substitute the values to find the percentage error in volume
Now, substituting the percentage error in radius into the equation:
\[ \text{Percentage error in volume} = 3 \times \left( \frac{0.1}{5.3} \times 100 \right) \]

### Step 7: Calculate the final value
Calculating the percentage error in volume:
\[ \text{Percentage error in volume} = 3 \times \left( \frac{0.1 \times 100}{5.3} \right) = 3 \times \left( \frac{10}{5.3} \right) \approx 3 \times 1.8868 \approx 5.66\% \]

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

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