Calculating Rectangle Area Percentage Increase
Let the original length of the rectangle be $L$ and the original width be $W$. The original area $A_{original}$ is given by:
$A_{original} = L \times W$
New Dimensions After Increase
Each side of the rectangle is increased by 10%. The new length $L_{new}$ and new width $W_{new}$ are:
- New Length: $L_{new} = L + (10\% \text{ of } L) = L + 0.10L = 1.10L$
- New Width: $W_{new} = W + (10\% \text{ of } W) = W + 0.10W = 1.10W$
Calculating New Area
The new area $A_{new}$ is calculated using the new dimensions:
$A_{new} = L_{new} \times W_{new} = (1.10L) \times (1.10W) = 1.21 \times (L \times W) = 1.21 LW$
Determining Percentage Increase in Area
The increase in area is the difference between the new area and the original area:
$ \text{Increase} = A_{new} - A_{original} = 1.21 LW - LW = 0.21 LW $
To find the percentage increase, we use the formula:
$ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Original Area}} \right) \times 100\% $
Substituting the values:
$ \text{Percentage Increase} = \left( \frac{0.21 LW}{LW} \right) \times 100\% = 0.21 \times 100\% = 21\% $
Using Successive Percentage Change Formula
Alternatively, for a rectangle where length and width are increased by $x\%$ and $y\%$ respectively, the percentage increase in area is given by $(x + y + \frac{xy}{100})\%$.
In this case, $x = 10\%$ and $y = 10\%$.
$ \text{Percentage Increase} = \left( 10 + 10 + \frac{10 \times 10}{100} \right)\% = \left( 20 + \frac{100}{100} \right)\% = (20 + 1)\% = 21\% $
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