The percentage error in the evaluation of the area of one surface of a rectangular sheet is 2.2%. The length and breadth are measured using a meter scale. If the breadth is 7.9 cm, calculate the length of the sheet?

Explanation

To solve the problem, we need to calculate the length of a rectangular sheet given the percentage error in the area and the breadth. Here’s a step-by-step solution:

### Step 1: Understand the relationship between area, length, and breadth
The area \( A \) of a rectangular sheet is given by the formula:
\[ A = L \times B \]
where \( L \) is the length and \( B \) is the breadth.

### Step 2: Use the formula for percentage error in area
The percentage error in the area can be expressed in terms of the percentage errors in length and breadth:
\[ \frac{\Delta A}{A} \times 100 = \frac{\Delta L}{L} \times 100 + \frac{\Delta B}{B} \times 100 \]
where \( \Delta A \), \( \Delta L \), and \( \Delta B \) are the absolute errors in area, length, and breadth respectively.

### Step 3: Substitute the known values
We know from the problem that:
- The percentage error in area \( \frac{\Delta A}{A} \times 100 = 2.2\% \)
- The breadth \( B = 7.9 \, \text{cm} \)
- The least count of the meter scale is \( 0.1 \, \text{cm} \), so we can assume \( \Delta L = \Delta B = 0.1 \, \text{cm} \).

Now we can rewrite the equation:
\[ 2.2 = \frac{\Delta L}{L} \times 100 + \frac{\Delta B}{B} \times 100 \]
Substituting \( \Delta L = 0.1 \) and \( \Delta B = 0.1 \):
\[ 2.2 = \frac{0.1}{L} \times 100 + \frac{0.1}{7.9} \times 100 \]

### Step 4: Calculate the contribution from breadth
First, calculate the second term:
\[ \frac{0.1}{7.9} \times 100 \approx 1.27 \]
So, we can rewrite the equation:
\[ 2.2 = \frac{10}{L} + 1.27 \]

### Step 5: Isolate the term involving length
Now, isolate \( \frac{10}{L} \):
\[ \frac{10}{L} = 2.2 - 1.27 \]
\[ \frac{10}{L} = 0.93 \]

### Step 6: Solve for length \( L \)
Now, we can solve for \( L \):
\[ L = \frac{10}{0.93} \approx 10.75 \, \text{cm} \]

### Step 7: Round off the answer
Rounding off to one decimal place, we get:
\[ L \approx 10.8 \, \text{cm} \]

### Final Answer
The length of the rectangular sheet is approximately \( 10.8 \, \text{cm} \).