The original width of a rectangular box is 20 cm. The box was made in such away that its length increased by 15% but the width decreased by 5%. As a result its area increased by `74 cm^(2)`. What is the length of box ?

Options

A. 20 cm
B. 60 cm
C. 40 cm
D. 30 cm

Answer

To find the length of the rectangular box, we will follow these steps:

### Step 1: Define the original dimensions
The original width (breadth) of the box is given as:
- Width (B) = 20 cm
- Let the original length be L cm.

### Step 2: Calculate the original area
The area (A) of a rectangle is calculated as:
\[ A = \text{Length} \times \text{Width} \]
Thus, the original area is:
\[ A_{\text{original}} = L \times 20 = 20L \, \text{cm}^2 \]

### Step 3: Determine the new dimensions after changes
The length increases by 15%, and the width decreases by 5%. Therefore, the new dimensions are:
- New Length (L') = \( L + 0.15L = 1.15L \)
- New Width (B') = \( 20 - 0.05 \times 20 = 20 - 1 = 19 \, \text{cm} \)

### Step 4: Calculate the new area
The new area (A') can be calculated as:
\[ A' = \text{New Length} \times \text{New Width} = 1.15L \times 19 \]

### Step 5: Set up the equation for area increase
According to the problem, the area increased by 74 cm². Therefore, we can write:
\[ A' - A_{\text{original}} = 74 \]
Substituting the values we have:
\[ (1.15L \times 19) - (20L) = 74 \]

### Step 6: Simplify the equation
Expanding the left side:
\[ 21.85L - 20L = 74 \]
This simplifies to:
\[ 1.85L = 74 \]

### Step 7: Solve for L
Now, we can solve for L:
\[ L = \frac{74}{1.85} \]
Calculating this gives:
\[ L \approx 40 \, \text{cm} \]

### Conclusion
Thus, the length of the box is:
\[ \boxed{40 \, \text{cm}} \]

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