The radius of a sphere is (2.6 `pm` 0.1) cm The percentage error u iis volume is

Options

A. `0.1/2.6xx100%`
B. `3xx0.1/2.6xx100%`
C. `0.1/(3xx2.6)xx100%`
D. `0.1/2.6%`

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: Identify the given values
The radius \( r \) of the sphere is given as:
\[ r = 2.6 \, \text{cm} \]
The uncertainty in the radius \( \Delta r \) is:
\[ \Delta r = 0.1 \, \text{cm} \]

### Step 2: Write 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 3: Determine how the volume depends on the radius
The volume \( V \) is directly proportional to \( r^3 \). Thus, the relative error in volume can be expressed in terms of the relative error in radius:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]

### Step 4: Calculate the relative error in radius
The relative error in radius is given by:
\[ \frac{\Delta r}{r} = \frac{0.1}{2.6} \]

### Step 5: Substitute the relative error into the volume formula
Now, substituting this into the volume error formula:
\[ \frac{\Delta V}{V} = 3 \left(\frac{0.1}{2.6}\right) \]

### Step 6: Calculate the relative error in volume
Calculating the above expression:
\[ \frac{\Delta V}{V} = 3 \times \frac{0.1}{2.6} = \frac{0.3}{2.6} \]

### Step 7: Convert to percentage error
To find the percentage error, multiply the relative error by 100:
\[ \text{Percentage Error} = \left(\frac{0.3}{2.6}\right) \times 100 \]

### Step 8: Perform the final calculation
Calculating this gives:
\[ \text{Percentage Error} \approx \left(0.1153846\right) \times 100 \approx 11.54\% \]

### Final Result
The percentage error in the volume of the sphere is approximately:
\[ 11.54\% \]