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

Question

Accepted Answer

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

1. **Initial Area Calculation:**
   - Let the original side length of the square be $ s $.
   - The original area of the square is $ s^2 $.

2. **New Side Length Calculation:**
   - When the side length is increased by 15%, the new side length becomes $ s + 0.15s = 1.15s $.

3. **New Area Calculation:**
   - The new area of the square with the increased side length is $ (1.15s)^2 = 1.3225s^2 $.

4. **Increase in Area:**
   - The increase in area is the new area minus the original area: $ 1.3225s^2 - s^2 = 0.3225s^2 $.

5. **Percentage Increase in Area:**
   - To find the percentage increase, divide the increase in area by the original area and multiply by 100:
   $$
   \text{Percentage Increase} = \left(\frac{0.3225s^2}{s^2}\right) \times 100 = 32.25\%
   $$

Thus, the increase in the area of the square is 32.25%, which corresponds to Option 4 (0.3225).

**Explanation of Incorrect Options:**

- **Option 1 (0.15):** This option suggests a 15% increase in area, which is incorrect because the percentage increase in area is not the same as the percentage increase in side length. The area increases by a factor of the square of the side length increase.

- **Option 2 (0.225):** This option suggests a 22.5% increase in area. This is incorrect because it does not account for the full effect of squaring the increased side length.

- **Option 3 (0.5625):** This option suggests a 56.25% increase in area, which is incorrect. This value is too high and does not accurately reflect the calculation of the area increase based on a 15% increase in side length.

Therefore, the correct answer is Option 4 (0.3225), as it accurately represents the percentage increase in the area of the square when each side is increased by 15%.