If `x:y:z=2:3:4`, what is the value of `((x^(2)+2y^(2)+z^(2))/(3x^(2)-2y^(2)+4z^(2)))`?

Options

A. `(3)/(4)`
B. `(21)/(29)`
C. `(5)/(6)`
D. `(19)/(29)`

Answer

To solve the problem given the ratio \( x:y:z = 2:3:4 \), we can express \( x \), \( y \), and \( z \) in terms of a common variable. Let's denote the common variable as \( k \).

### Step 1: Express \( x \), \( y \), and \( z \) in terms of \( k \)
From the ratio:
- \( x = 2k \)
- \( y = 3k \)
- \( z = 4k \)

### Step 2: Substitute \( x \), \( y \), and \( z \) into the expression
We need to evaluate:
\[ \frac{x^2 + 2y^2 + z^2}{3x^2 - 2y^2 + 4z^2} \]

Substituting the values of \( x \), \( y \), and \( z \):
\[ x^2 = (2k)^2 = 4k^2 \]
\[ y^2 = (3k)^2 = 9k^2 \]
\[ z^2 = (4k)^2 = 16k^2 \]

Now substituting these into the expression:
\[ \text{Numerator: } x^2 + 2y^2 + z^2 = 4k^2 + 2(9k^2) + 16k^2 \]
\[ = 4k^2 + 18k^2 + 16k^2 = 38k^2 \]

\[ \text{Denominator: } 3x^2 - 2y^2 + 4z^2 = 3(4k^2) - 2(9k^2) + 4(16k^2) \]
\[ = 12k^2 - 18k^2 + 64k^2 = 58k^2 \]

### Step 3: Simplify the expression
Now we can simplify the fraction:
\[ \frac{38k^2}{58k^2} \]
The \( k^2 \) terms cancel out:
\[ = \frac{38}{58} \]

### Step 4: Reduce the fraction
To reduce \( \frac{38}{58} \):
\[ = \frac{19}{29} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{19}{29} \]