If a = 15, b = 12 and c= 9, find the value of `sqrt(((2a+2b)(2c+a))/(2(a-b)(2b+c)))`

Options

A. 10
B. 6
C. 9
D. 3

Answer

To solve the expression \( \sqrt{\frac{(2a + 2b)(2c + a)}{2(a - b)(2b + c)}} \) given \( a = 15 \), \( b = 12 \), and \( c = 9 \), we can follow these steps:

### Step 1: Substitute the values of a, b, and c into the expression.

Given:
- \( a = 15 \)
- \( b = 12 \)
- \( c = 9 \)

Substituting these values into the expression:

\[ \sqrt{\frac{(2(15) + 2(12))(2(9) + 15)}{2(15 - 12)(2(12) + 9)}} \]

### Step 2: Calculate the components inside the parentheses.

Calculate \( 2a + 2b \):

\[ 2(15) + 2(12) = 30 + 24 = 54 \]

Calculate \( 2c + a \):

\[ 2(9) + 15 = 18 + 15 = 33 \]

Calculate \( a - b \):

\[ 15 - 12 = 3 \]

Calculate \( 2b + c \):

\[ 2(12) + 9 = 24 + 9 = 33 \]

### Step 3: Substitute these results back into the expression.

Now we can substitute these results into the expression:

\[ \sqrt{\frac{(54)(33)}{2(3)(33)}} \]

### Step 4: Simplify the expression.

First, simplify the denominator:

\[ 2(3)(33) = 198 \]

Now, the expression becomes:

\[ \sqrt{\frac{54 \times 33}{198}} \]

### Step 5: Simplify the fraction.

Notice that \( 198 = 6 \times 33 \). Therefore, we can simplify:

\[ \frac{54 \times 33}{198} = \frac{54 \times 33}{6 \times 33} = \frac{54}{6} = 9 \]

### Step 6: Take the square root.

Now we take the square root of 9:

\[ \sqrt{9} = 3 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{3} \]