The radius of a sphere is `(4.8 +- 0.1)`cm` The perecentage error in its volume is

Answer

To find the percentage error in the volume of a sphere given its 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 error in the radius
The radius is given as \( r = 4.8 \pm 0.1 \) cm. Here, \( \Delta R \) (the absolute error in the radius) is \( 0.1 \) cm.

### Step 3: Calculate the percentage error in the volume
The percentage error in volume can be calculated using the formula:
\[ \text{Percentage Error in } V = \left( \frac{\Delta V}{V} \right) \times 100 \]
Using the relationship between the errors, we have:
\[ \frac{\Delta V}{V} = 3 \times \frac{\Delta R}{R} \]
This is because the volume depends on the cube of the radius.

### Step 4: Substitute the values into the formula
Substituting \( \Delta R = 0.1 \) cm and \( R = 4.8 \) cm, we get:
\[ \frac{\Delta V}{V} = 3 \times \frac{0.1}{4.8} \]

### Step 5: Simplify the expression
Calculating the fraction:
\[ \frac{0.1}{4.8} = \frac{1}{48} \]
Thus,
\[ \frac{\Delta V}{V} = 3 \times \frac{1}{48} = \frac{3}{48} \]

### Step 6: Convert to percentage
Now, to find the percentage error:
\[ \text{Percentage Error in } V = \left( \frac{3}{48} \right) \times 100 \]
Calculating this gives:
\[ \text{Percentage Error in } V = \frac{300}{48} \approx 6.25\% \]

### Final Answer
The percentage error in the volume of the sphere is approximately **6.25%**.

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