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 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.
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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.
Updated On: Apr 4, 2025
Perimeter increases by 10%.
Length of the diagonals increases by 10%.
Area increases by 10%.
- The rectangle becomes a square.
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The Correct Option is C
Solution and Explanation
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 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%.
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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