The radius of a sphere is measured to be `(2.1+-0.5)` cm. Calculate its surface area with error limits .

Explanation

To solve the problem of calculating the surface area of a sphere given the radius and its error limits, we can follow these steps:

### Step 1: Identify the given values
The radius of the sphere is given as:
\[ r = 2.1 \pm 0.5 \text{ cm} \]
This means the radius \( r \) is \( 2.1 \) cm with a possible error of \( \pm 0.5 \) cm.

### Step 2: Write the formula for the surface area of a sphere
The formula for the surface area \( A \) of a sphere is:
\[ A = 4 \pi r^2 \]

### Step 3: Calculate the surface area using the nominal value of the radius
Substituting the nominal value of the radius into the formula:
\[ A = 4 \pi (2.1)^2 \]
Calculating \( (2.1)^2 \):
\[ (2.1)^2 = 4.41 \]
Now substituting back into the surface area formula:
\[ A = 4 \times 3.14 \times 4.41 \]
Calculating this gives:
\[ A \approx 55.4 \text{ cm}^2 \]

### Step 4: Calculate the error in the surface area
To find the error in the surface area, we use the formula for relative error:
\[ \frac{\Delta A}{A} = 2 \frac{\Delta r}{r} \]
Where:
- \( \Delta A \) is the error in the surface area,
- \( \Delta r \) is the error in the radius,
- \( r \) is the nominal radius.

Substituting the known values:
\[ \Delta r = 0.5 \text{ cm} \]
\[ r = 2.1 \text{ cm} \]
So,
\[ \frac{\Delta A}{A} = 2 \frac{0.5}{2.1} \]

Calculating this:
\[ \frac{\Delta A}{A} \approx 2 \times 0.2381 \approx 0.4762 \]

### Step 5: Calculate the absolute error in the surface area
Now we can find \( \Delta A \):
\[ \Delta A = A \times \frac{\Delta A}{A} \]
Substituting the values:
\[ \Delta A \approx 55.4 \times 0.4762 \approx 26.4 \text{ cm}^2 \]

### Step 6: Present the final result
The surface area of the sphere with error limits is:
\[ A = 55.4 \pm 26.4 \text{ cm}^2 \]

### Summary
The surface area of the sphere is \( 55.4 \text{ cm}^2 \) with an error limit of \( \pm 26.4 \text{ cm}^2 \).

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