The radius of a sphere is measured to be `(2.1 +- 0.02) cm`. Calculate its surface area with error limits.

Explanation

To calculate the surface area of a sphere given the radius and its uncertainty, we can follow these steps:

### Step 1: Write down the formula for the surface area of a sphere.
The surface area \( S \) of a sphere is given by the formula:
\[ S = 4 \pi r^2 \]

### Step 2: Substitute the radius into the formula.
Given the radius \( r = 2.1 \, \text{cm} \), we can substitute this value into the formula:
\[ S = 4 \pi (2.1)^2 \]

### Step 3: Calculate \( (2.1)^2 \).
First, we calculate \( (2.1)^2 \):
\[ (2.1)^2 = 4.41 \]

### Step 4: Calculate the surface area.
Now substitute \( (2.1)^2 \) back into the surface area formula:
\[ S = 4 \pi (4.41) \approx 4 \times 3.14 \times 4.41 \]
Calculating this gives:
\[ S \approx 4 \times 3.14 \times 4.41 \approx 55.4 \, \text{cm}^2 \]

### Step 5: Calculate the error in the surface area.
To find the error limit, we need to consider the uncertainty in the radius. The uncertainty in the radius \( \Delta r = 0.02 \, \text{cm} \).

Using the formula for the error in the surface area, we have:
\[ \Delta S = \left| \frac{dS}{dr} \right| \Delta r \]
Where \( \frac{dS}{dr} \) is the derivative of the surface area with respect to the radius.

### Step 6: Calculate \( \frac{dS}{dr} \).
From the surface area formula:
\[ S = 4 \pi r^2 \]
Taking the derivative with respect to \( r \):
\[ \frac{dS}{dr} = 8 \pi r \]

### Step 7: Substitute the radius into the derivative.
Substituting \( r = 2.1 \, \text{cm} \):
\[ \frac{dS}{dr} = 8 \pi (2.1) \approx 8 \times 3.14 \times 2.1 \approx 52.8 \]

### Step 8: Calculate the error in the surface area.
Now we can calculate \( \Delta S \):
\[ \Delta S = 52.8 \times 0.02 \approx 1.056 \]

### Step 9: Write the final result.
Now we can express the surface area with its error limits:
\[ S = 55.4 \pm 1.056 \, \text{cm}^2 \]

### Final Answer:
The surface area of the sphere is approximately:
\[ S = 55.4 \pm 1.1 \, \text{cm}^2 \]