We are given the system of equations:

$3a = 4b = 6c$

and

$a + b + c = 27\sqrt{29}$

Step 1: Express $a$, $b$, and $c$ in terms of one variable

From the equation $3a = 4b = 6c$, we can introduce a variable $k$ such that:

$3a = 4b = 6c = k$

This gives us the following expressions:

$a = \frac{k}{3}, \quad b = \frac{k}{4}, \quad c = \frac{k}{6}$

Step 2: Use the equation $a + b + c = 27\sqrt{29}$

Substitute $a = \frac{k}{3}$, $b = \frac{k}{4}$, and $c = \frac{k}{6}$ into the equation $a + b + c = 27\sqrt{29}$:

$\frac{k}{3} + \frac{k}{4} + \frac{k}{6} = 27\sqrt{29}$

To add these fractions, find the least common denominator (LCD) of 3, 4, and 6, which is 12:

$\frac{k}{3} = \frac{4k}{12}, \quad \frac{k}{4} = \frac{3k}{12}, \quad \frac{k}{6} = \frac{2k}{12}$

Now add the fractions:

$\frac{4k}{12} + \frac{3k}{12} + \frac{2k}{12} = \frac{9k}{12} = \frac{3k}{4}$

So, we have:

$\frac{3k}{4} = 27\sqrt{29}$

Step 3: Solve for $k$

Multiply both sides by 4:

$3k = 108\sqrt{29}$

Now divide by 3:

$k = 36\sqrt{29}$

Step 4: Find $\sqrt{a^2 + b^2 + c^2}$

Now, we need to compute $\sqrt{a^2 + b^2 + c^2}$. Using the expressions for $a$, $b$, and $c$:

$a = \frac{k}{3} = \frac{36\sqrt{29}}{3} = 12\sqrt{29}$ $b = \frac{k}{4} = \frac{36\sqrt{29}}{4} = 9\sqrt{29}$ $c = \frac{k}{6} = \frac{36\sqrt{29}}{6} = 6\sqrt{29}$

Now compute $a^2 + b^2 + c^2$:

$a^2 = (12\sqrt{29})^2 = 144 \times 29 = 4176$ $b^2 = (9\sqrt{29})^2 = 81 \times 29 = 2349$ $c^2 = (6\sqrt{29})^2 = 36 \times 29 = 1044$

Now add these:

$a^2 + b^2 + c^2 = 4176 + 2349 + 1044 = 7569$

Step 5: Take the square root

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

Thus, the value of $\sqrt{a^2 + b^2 + c^2}$ is $87$.

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Would you like any further details or clarifications on the steps?

Related questions:

1. How would the solution change if the values of $a + b + c$ were different?
2. How can this method be applied to solve other systems of proportional equations?
3. What is the significance of introducing the variable $k$ in solving this type of equation?
4. Can this approach be generalized for $ma = nb = pc$ type of systems?
5. How do different numerical values of $a$, $b$, and $c$ affect the value of $\sqrt{a^2 + b^2 + c^2}$?

Tip:

When dealing with proportional equations, using a common variable like $k$ can simplify the process of solving for multiple unknowns at once.