Question

A rectangle has a length \(L\) and a width \(W\), where \(L>W\). If the width, \(W\), is increased by 10%, which one of the following statements is correct for all values of \(L\) and \(W\)? 
Select the most appropriate option to complete the above sentence.

• A. Perimeter increases by 10%.
• B. Length of the diagonals increases by 10%.
• C. Area increases by 10%.
• D. The rectangle becomes a square.

Hint:

When a single dimension of a rectangle (such as width) is increased by a certain percentage, the area of the rectangle will increase by the same percentage, as long as the other dimension remains unchanged.

Answer 1

The Correct Option is C

Step 1: Understanding the effects of increasing the width of a rectangle by 10%
The dimensions of the rectangle are \( L \) (length) and \( W \) (width), with \( L > W \). When the width \( W \) is increased by 10%, the new width becomes \( W' = 1.1W \), while the length \( L \) remains the same.

Step 2: Analyzing the impact on each option

Option (A): Perimeter increases by 10%
The perimeter of a rectangle is given by the formula:
\[ P = 2(L + W) \]
When the width increases by 10%, the new perimeter becomes:
\[ P' = 2(L + 1.1W) \]
This is not exactly a 10% increase. The increase in perimeter is not proportional to the increase in width. Therefore, this option is incorrect.

Option (B): Length of the diagonals increases by 10%
The diagonal \( d \) of a rectangle is given by the Pythagorean theorem:
\[ d = \sqrt{L^2 + W^2} \]
When the width increases by 10%, the new diagonal is:
\[ d' = \sqrt{L^2 + (1.1W)^2} \]
This increase is not guaranteed to be exactly 10%. The length of the diagonal increases, but not necessarily by 10%. Therefore, this option is incorrect.

Option (C): Area increases by 10%
The area \( A \) of a rectangle is:
\[ A = L \times W \]
After increasing the width by 10%, the new area is:
\[ A' = L \times 1.1W = 1.1 \times L \times W \]
This shows that the area increases by exactly 10%. Therefore, this option is correct.

Option (D): The rectangle becomes a square
A rectangle becomes a square only if the length and width are equal. Since only the width is increased by 10% and \( L > W \), this option is incorrect.

Therefore, the correct answer is Option (C): Area increases by 10%.