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 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.
Show Hint
Perimeter increases by 10%.
Length of the diagonals increases by 10%.
Area increases by 10%.
- The rectangle becomes a square.
The Correct Option is C
Solution and Explanation
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 it is not necessarily a 10% increase. Therefore, this option is incorrect.
Option (C) Area increases by 10%:
The area \( A \) of a rectangle is given by: \[ A = L \times W \] 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 10%, as the new area is 1.1 times the original area. 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%, the rectangle does not become a square. Therefore, this option is incorrect.
Therefore, the correct answer is Option (C) Area increases by 10%.
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