If `x` is added to each of 14, 12, 34 and 30, the numbers so obtained, in this order, are in proportion. What is the value of `sqrt(12x+9 ` ?

Options

A. 9
B. 11
C. 13
D. 8

Answer

To solve the problem step by step, we need to find the value of \( \sqrt{12x + 9} \) given that if \( x \) is added to each of the numbers 14, 12, 34, and 30, the resulting numbers are in proportion.

### Step 1: Set up the proportion
When \( x \) is added to each number, we have:
- First term: \( 14 + x \)
- Second term: \( 12 + x \)
- Third term: \( 34 + x \)
- Fourth term: \( 30 + x \)

These numbers are in proportion, which means:
\[ \frac{14 + x}{12 + x} = \frac{34 + x}{30 + x} \]

### Step 2: Cross-multiply the proportions
Cross-multiplying gives us:
\[ (14 + x)(30 + x) = (12 + x)(34 + x) \]

### Step 3: Expand both sides
Expanding both sides:
- Left side:
\[ 14 \cdot 30 + 14x + 30x + x^2 = 420 + 44x + x^2 \]
- Right side:
\[ 12 \cdot 34 + 12x + 34x + x^2 = 408 + 46x + x^2 \]

### Step 4: Set the equation
Now we set the expanded forms equal to each other:
\[ 420 + 44x + x^2 = 408 + 46x + x^2 \]

### Step 5: Simplify the equation
Subtract \( x^2 \) from both sides:
\[ 420 + 44x = 408 + 46x \]
Now, subtract \( 44x \) from both sides:
\[ 420 = 408 + 2x \]
Subtract 408 from both sides:
\[ 12 = 2x \]
Now, divide by 2:
\[ x = 6 \]

### Step 6: Substitute \( x \) into \( 12x + 9 \)
Now we need to find \( \sqrt{12x + 9} \):
\[ 12x + 9 = 12(6) + 9 = 72 + 9 = 81 \]

### Step 7: Calculate the square root
Finally, we take the square root:
\[ \sqrt{81} = 9 \]

### Final Answer
The value of \( \sqrt{12x + 9} \) is \( 9 \).

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