The sides of a rectangle are `(10.5 +- 0.2) cm and ( 5.2 +- 0.1 ) cm`. Calculate its perimeter with error limits .

Explanation

To calculate the perimeter of a rectangle with given side lengths and their uncertainties, we can follow these steps:

### Step 1: Identify the given values
The lengths of the sides of the rectangle are:
- Length (l) = \(10.5 \, \text{cm} \pm 0.2 \, \text{cm}\)
- Breadth (b) = \(5.2 \, \text{cm} \pm 0.1 \, \text{cm}\)

### Step 2: Write the formula for the perimeter
The formula for the perimeter (P) of a rectangle is given by:
\[ P = 2 \times (l + b) \]

### Step 3: Calculate the perimeter
Substituting the values of l and b into the formula:
\[ P = 2 \times (10.5 \, \text{cm} + 5.2 \, \text{cm}) \]
Calculating the sum inside the parentheses:
\[ P = 2 \times (15.7 \, \text{cm}) \]
Now, multiplying by 2:
\[ P = 31.4 \, \text{cm} \]

### Step 4: Calculate the error in the perimeter
To find the error in the perimeter (\(\Delta P\)), we use the formula for the propagation of uncertainty:
\[ \Delta P = 2 \times (\Delta l + \Delta b) \]
Where:
- \(\Delta l = 0.2 \, \text{cm}\)
- \(\Delta b = 0.1 \, \text{cm}\)

Substituting the values:
\[ \Delta P = 2 \times (0.2 \, \text{cm} + 0.1 \, \text{cm}) \]
Calculating the sum:
\[ \Delta P = 2 \times (0.3 \, \text{cm}) \]
Now, multiplying by 2:
\[ \Delta P = 0.6 \, \text{cm} \]

### Step 5: Write the final result
The perimeter of the rectangle with its error limits is:
\[ P = 31.4 \, \text{cm} \pm 0.6 \, \text{cm} \]

### Final Answer:
The perimeter of the rectangle is \(31.4 \, \text{cm} \pm 0.6 \, \text{cm}\).

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