If `3a=4b=6c` and `a+b+c=27sqrt(29)` then `sqrt(a^(2)+b^(2)+c^(2))` is equal to

Options

A. `87`
B. `3sqrt(29)`
C. `82`
D. `83

Answer

To solve the problem, we start with the given equations:

1. **Given**: \( 3a = 4b = 6c \)
2. **Given**: \( a + b + c = 27\sqrt{29} \)

### Step 1: Express \( a \), \( b \), and \( c \) in terms of a common variable

From the equation \( 3a = 4b \), we can express \( a \) in terms of \( b \):

\[ a = \frac{4}{3}b \]

From the equation \( 4b = 6c \), we can express \( b \) in terms of \( c \):

\[ b = \frac{6}{4}c = \frac{3}{2}c \]

Now we can express \( a \) in terms of \( c \):

\[ a = \frac{4}{3} \left(\frac{3}{2}c\right) = 2c \]

So, we have:

\[ a = 2c, \quad b = \frac{3}{2}c, \quad c = c \]

### Step 2: Substitute \( a \), \( b \), and \( c \) into the sum equation

Now substitute \( a \), \( b \), and \( c \) into the equation \( a + b + c = 27\sqrt{29} \):

\[ 2c + \frac{3}{2}c + c = 27\sqrt{29} \]

Combining the terms:

\[ 2c + \frac{3}{2}c + 1c = 2c + 1c + \frac{3}{2}c = \left(2 + 1 + \frac{3}{2}\right)c = \left(3 + \frac{3}{2}\right)c = \frac{6}{2}c + \frac{3}{2}c = \frac{9}{2}c \]

Thus, we have:

\[ \frac{9}{2}c = 27\sqrt{29} \]

### Step 3: Solve for \( c \)

To find \( c \), multiply both sides by \( \frac{2}{9} \):

\[ c = 27\sqrt{29} \cdot \frac{2}{9} = 6\sqrt{29} \]

### Step 4: Find \( a \) and \( b \)

Now substitute \( c \) back to find \( a \) and \( b \):

\[ a = 2c = 2 \cdot 6\sqrt{29} = 12\sqrt{29} \]

\[ b = \frac{3}{2}c = \frac{3}{2} \cdot 6\sqrt{29} = 9\sqrt{29} \]

### Step 5: Calculate \( \sqrt{a^2 + b^2 + c^2} \)

Now we need to calculate \( \sqrt{a^2 + b^2 + c^2} \):

\[ a^2 = (12\sqrt{29})^2 = 144 \cdot 29 = 4176 \]

\[ b^2 = (9\sqrt{29})^2 = 81 \cdot 29 = 2349 \]

\[ c^2 = (6\sqrt{29})^2 = 36 \cdot 29 = 1044 \]

Now sum these values:

\[ a^2 + b^2 + c^2 = 4176 + 2349 + 1044 = 7569 \]

### Step 6: Take the square root

Now take the square root:

\[ \sqrt{a^2 + b^2 + c^2} = \sqrt{7569} = 87 \]

Thus, the final answer is:

\[ \sqrt{a^2 + b^2 + c^2} = 87 \]