If the length of each side of a square is increased by 15% the the increase percent in its area is

Options

A. `15%`
B. `22.5%`
C. `56.25%`
D. `32.25%`

Answer

To find the increase in the area of a square when the length of each side is increased by 15%, we can follow these steps:

### Step 1: Understand the original area of the square
Let the original length of each side of the square be \( s \). The area \( A \) of the square is given by the formula:
\[ A = s^2 \]

### Step 2: Calculate the new length of each side after the increase
If the length of each side is increased by 15%, the new length \( s' \) can be calculated as:
\[ s' = s + 0.15s = 1.15s \]

### Step 3: Calculate the new area of the square
The new area \( A' \) of the square with the new side length is:
\[ A' = (s')^2 = (1.15s)^2 \]
Expanding this, we get:
\[ A' = 1.3225s^2 \]

### Step 4: Calculate the increase in area
The increase in area \( \Delta A \) is given by:
\[ \Delta A = A' - A = 1.3225s^2 - s^2 = (1.3225 - 1)s^2 = 0.3225s^2 \]

### Step 5: Calculate the percentage increase in area
To find the percentage increase in area, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta A}{A} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left( \frac{0.3225s^2}{s^2} \right) \times 100 = 0.3225 \times 100 = 32.25\% \]

### Final Answer
The increase in area is **32.25%**.

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