3. The dimensions of a rectangle are 40 cm and 45 cm . If there is an error of $$ 5 \% $$ in the dimensions find the percentage error in calculating area of the rectangle. ( 3 mks )

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To find the percentage error in calculating the area of the rectangle due to the error in its dimensions, we can follow these steps:

1. **Identify the dimensions of the rectangle:**
   - Length $$ L = 40 $$ cm
   - Width $$ W = 45 $$ cm

2. **Calculate the area of the rectangle without error:**
   $$
   A = L \times W = 40 \times 45
   $$

3. **Calculate the percentage error in the dimensions:**
   - Given that there is a $$ 5\% $$ error in both dimensions, we can express the errors in length and width as:
     - Error in Length $$ = 0.05 \times L $$
     - Error in Width $$ = 0.05 \times W $$

4. **Calculate the new dimensions considering the error:**
   - New Length $$ L' = L + \text{Error in Length} = L + 0.05L = 1.05L $$
   - New Width $$ W' = W + \text{Error in Width} = W + 0.05W = 1.05W $$

5. **Calculate the new area with the erroneous dimensions:**
   $$
   A' = L' \times W' = (1.05L) \times (1.05W) = 1.05^2 \times (L \times W)
   $$

6. **Calculate the percentage error in the area:**
   - The percentage error in the area can be calculated using the formula:
   $$
   \text{Percentage Error in Area} = \left( \frac{A' - A}{A} \right) \times 100\%
   $$

Now, let's perform the calculations step by step.

### Step 2: Calculate the area of the rectangle without error
$$
A = 40 \times 45
$$

### Step 4: Calculate the new dimensions
$$
L' = 1.05 \times 40
$$
$$
W' = 1.05 \times 45
$$

### Step 5: Calculate the new area
$$
A' = L' \times W'
$$

### Step 6: Calculate the percentage error in the area
$$
\text{Percentage Error in Area} = \left( \frac{A' - A}{A} \right) \times 100\%
$$

Now, I will perform the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$40\times 45$$
- step1: Multiply the numbers:
  $$1800$$
Calculate or simplify the expression $$ 1.05 * 45 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1.05\times 45$$
- step1: Multiply the numbers:
  $$47.25$$
Calculate or simplify the expression $$ 1.05 * 40 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1.05\times 40$$
- step1: Multiply the numbers:
  $$42$$
Calculate or simplify the expression $$ 42 * 47.25 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$42\times 47.25$$
- step1: Multiply the numbers:
  $$1984.5$$
Calculate or simplify the expression $$ (1984.5 - 1800) / 1800 * 100 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(1984.5-1800\right)}{1800}\times 100$$
- step1: Remove the parentheses:
  $$\frac{1984.5-1800}{1800}\times 100$$
- step2: Subtract the numbers:
  $$\frac{184.5}{1800}\times 100$$
- step3: Divide the terms:
  $$\frac{41}{400}\times 100$$
- step4: Reduce the numbers:
  $$\frac{41}{4}\times 1$$
- step5: Multiply:
  $$\frac{41}{4}$$
Let's summarize the calculations step by step:

1. **Area of the rectangle without error:**
   $$
   A = 40 \times 45 = 1800 \text{ cm}^2
   $$

2. **New dimensions considering the $$ 5\% $$ error:**
   - New Length:
   $$
   L' = 1.05 \times 40 = 42 \text{ cm}
   $$
   - New Width:
   $$
   W' = 1.05 \times 45 = 47.25 \text{ cm}
   $$

3. **New area with the erroneous dimensions:**
   $$
   A' = 42 \times 47.25 = 1984.5 \text{ cm}^2
   $$

4. **Percentage error in the area:**
   $$
   \text{Percentage Error in Area} = \left( \frac{1984.5 - 1800}{1800} \right) \times 100\% = \left( \frac{184.5}{1800} \right) \times 100\% \approx 10.25\%
   $$

Thus, the percentage error in calculating the area of the rectangle is approximately **10.25%**.
The percentage error in calculating the area of the rectangle is approximately 10.25%.

3. The dimensions of a rectangle are 40 cm and 45 cm . If there is an error of \( 5 \% \) in the dimensions find the percentage error in calculating area of the rectangle. ( 3 mks )

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