The radius of a sphere is measured to be `(1.2 +- 0.2)cm`. Calculate its volume with error limits.

Explanation

To calculate the volume of a sphere with a given radius and its associated error limits, 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: Substitute the given radius into the formula
The radius is given as \( r = 1.2 \, \text{cm} \). We can substitute this value into the volume formula:
\[ V = \frac{4}{3} \pi (1.2)^3 \]

### Step 3: Calculate \( (1.2)^3 \)
Calculating \( (1.2)^3 \):
\[ (1.2)^3 = 1.2 \times 1.2 \times 1.2 = 1.728 \, \text{cm}^3 \]

### Step 4: Calculate the volume
Now, substituting \( (1.2)^3 \) back into the volume formula:
\[ V = \frac{4}{3} \pi (1.728) \]
Using \( \pi \approx 3.14 \):
\[ V \approx \frac{4}{3} \times 3.14 \times 1.728 \]
Calculating this step-by-step:
\[ \frac{4}{3} \times 3.14 \approx 4.18667 \]
Now multiply by \( 1.728 \):
\[ V \approx 4.18667 \times 1.728 \approx 7.224 \, \text{cm}^3 \]

### Step 5: Calculate the error in the volume
The error in the radius is given as \( \Delta r = 0.2 \, \text{cm} \). To find the error in volume \( \Delta V \), we use the formula for the propagation of uncertainty:
\[ \frac{\Delta V}{V} = 3 \frac{\Delta r}{r} \]
Substituting the values:
\[ \frac{\Delta V}{7.224} = 3 \frac{0.2}{1.2} \]
Calculating \( \frac{0.2}{1.2} = \frac{1}{6} \approx 0.16667 \):
\[ \frac{\Delta V}{7.224} = 3 \times 0.16667 \approx 0.5 \]
Now, multiplying both sides by \( 7.224 \):
\[ \Delta V \approx 0.5 \times 7.224 \approx 3.612 \, \text{cm}^3 \]

### Step 6: Write the final result
The volume of the sphere with error limits is:
\[ V = 7.224 \pm 3.612 \, \text{cm}^3 \]

### Final Answer
Thus, the final answer is:
\[ V = 7.2 \pm 3.6 \, \text{cm}^3 \]