If `9a^(2) + 16b^(2) + c^(2) + 25 = 24, (a+b)`, then the value of `(3a + 4b + 5c)` is :

Options

A. 9
B. 6
C. 7
D. 10

Answer

To solve the equation \( 9a^2 + 16b^2 + c^2 + 25 = 24(a + b) \) and find the value of \( 3a + 4b + 5c \), we can follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to set it to zero:
\[ 9a^2 + 16b^2 + c^2 + 25 - 24a - 24b = 0 \]

### Step 2: Complete the square for \( a \)
To complete the square for the \( a \) terms:
\[ 9a^2 - 24a = 9(a^2 - \frac{24}{9}a) = 9\left(a^2 - \frac{8}{3}a\right) \]
Now, take half of \(-\frac{8}{3}\), square it, and add/subtract it inside the bracket:
\[ = 9\left(a^2 - \frac{8}{3}a + \frac{16}{9} - \frac{16}{9}\right) = 9\left(\left(a - \frac{4}{3}\right)^2 - \frac{16}{9}\right) \]
This simplifies to:
\[ = 9\left(a - \frac{4}{3}\right)^2 - 16 \]

### Step 3: Complete the square for \( b \)
Now, complete the square for the \( b \) terms:
\[ 16b^2 - 24b = 16(b^2 - \frac{24}{16}b) = 16\left(b^2 - \frac{3}{2}b\right) \]
Again, take half of \(-\frac{3}{2}\), square it, and add/subtract:
\[ = 16\left(b^2 - \frac{3}{2}b + \frac{9}{16} - \frac{9}{16}\right) = 16\left(\left(b - \frac{3}{4}\right)^2 - \frac{9}{16}\right) \]
This simplifies to:
\[ = 16\left(b - \frac{3}{4}\right)^2 - 9 \]

### Step 4: Substitute back into the equation
Now substitute these completed squares back into the equation:
\[ 9\left(a - \frac{4}{3}\right)^2 - 16 + 16\left(b - \frac{3}{4}\right)^2 - 9 + c^2 + 25 = 0 \]
Combine the constants:
\[ 9\left(a - \frac{4}{3}\right)^2 + 16\left(b - \frac{3}{4}\right)^2 + c^2 = 0 \]
Since squares are always non-negative, the only solution occurs when each square is zero:
\[ 9\left(a - \frac{4}{3}\right)^2 = 0 \implies a = \frac{4}{3} \]
\[ 16\left(b - \frac{3}{4}\right)^2 = 0 \implies b = \frac{3}{4} \]
\[ c^2 = 0 \implies c = 0 \]

### Step 5: Calculate \( 3a + 4b + 5c \)
Now substitute \( a \), \( b \), and \( c \) into \( 3a + 4b + 5c \):
\[ 3a + 4b + 5c = 3\left(\frac{4}{3}\right) + 4\left(\frac{3}{4}\right) + 5(0) \]
Calculating this gives:
\[ = 4 + 3 + 0 = 7 \]

### Final Answer
Thus, the value of \( 3a + 4b + 5c \) is \( \boxed{7} \).