Due to increase of `k%` in the side, the area of a square increases by `69%`. What is the value of k ?

Options

A. `30`
B. `33`
C. `34.4`
D. `35`

Answer

To solve the problem, we need to find the value of \( k \) such that when the side of a square is increased by \( k\% \), the area of the square increases by \( 69\% \).

### Step 1: Understand the relationship between side and area of a square
The area \( A \) of a square is given by the formula:
\[ A = \text{side}^2 \]
If the original side length is \( s \), then the original area is:
\[ A_1 = s^2 \]

### Step 2: Calculate the new side length after the increase
If the side is increased by \( k\% \), the new side length \( s' \) can be expressed as:
\[ s' = s + \frac{k}{100} \cdot s = s \left(1 + \frac{k}{100}\right) \]

### Step 3: Calculate the new area with the new side length
The new area \( A_2 \) after the increase in side length is:
\[ A_2 = (s')^2 = \left(s \left(1 + \frac{k}{100}\right)\right)^2 = s^2 \left(1 + \frac{k}{100}\right)^2 \]

### Step 4: Relate the new area to the increase in area
According to the problem, the area increases by \( 69\% \). Therefore:
\[ A_2 = A_1 + 0.69 A_1 = 1.69 A_1 \]
Substituting \( A_1 = s^2 \):
\[ A_2 = 1.69 s^2 \]

### Step 5: Set the equations equal to each other
From the expressions for \( A_2 \):
\[ s^2 \left(1 + \frac{k}{100}\right)^2 = 1.69 s^2 \]
Since \( s^2 \) is common on both sides, we can cancel it out (assuming \( s \neq 0 \)):
\[ \left(1 + \frac{k}{100}\right)^2 = 1.69 \]

### Step 6: Take the square root of both sides
Taking the square root:
\[ 1 + \frac{k}{100} = \sqrt{1.69} \]
Calculating the square root:
\[ \sqrt{1.69} = 1.3 \]
So we have:
\[ 1 + \frac{k}{100} = 1.3 \]

### Step 7: Solve for \( k \)
Subtracting 1 from both sides:
\[ \frac{k}{100} = 1.3 - 1 = 0.3 \]
Multiplying both sides by 100:
\[ k = 0.3 \times 100 = 30 \]

### Final Answer
The value of \( k \) is \( 30\% \).

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