Answer
The percentage error in calculating the area of the rectangle is approximately 10.25%.
Solution
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%**.
Answered by UpStudy AI and reviewed by a Professional Tutor
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