If ` 1 . 5 x = 0.04 y` ,t hen the value of ` ( y^(2) - x^(2))/( y^(2) + 2xy + x^(2)) ` is

Options

A. A)`(730)/(77)`
B. B)`(73)/(77)`
C. C)`(73)/(770)`
D. D)`(74)/(77)`

Answer

To solve the equation \( 1.5x = 0.04y \) and find the value of \( \frac{y^2 - x^2}{y^2 + 2xy + x^2} \), we can follow these steps:

### Step 1: Find the relationship between \( x \) and \( y \)

Starting from the equation:
\[ 1.5x = 0.04y \]
We can rearrange this to express \( \frac{x}{y} \):
\[ \frac{x}{y} = \frac{0.04}{1.5} \]

### Step 2: Simplify the fraction

To simplify \( \frac{0.04}{1.5} \), we can eliminate the decimal points:
\[ \frac{0.04}{1.5} = \frac{4}{150} = \frac{2}{75} \]
Thus, we have:
\[ \frac{x}{y} = \frac{2}{75} \]

### Step 3: Assign values to \( x \) and \( y \)

From \( \frac{x}{y} = \frac{2}{75} \), we can assign:
\[ x = 2 \quad \text{and} \quad y = 75 \]

### Step 4: Substitute \( x \) and \( y \) into the expression

Now we need to find the value of:
\[ \frac{y^2 - x^2}{y^2 + 2xy + x^2} \]

Calculating \( y^2 - x^2 \):
\[ y^2 - x^2 = 75^2 - 2^2 = 5625 - 4 = 5621 \]

Calculating \( y^2 + 2xy + x^2 \):
\[ y^2 + 2xy + x^2 = 75^2 + 2 \cdot 75 \cdot 2 + 2^2 = 5625 + 300 + 4 = 5929 \]

### Step 5: Substitute the results into the expression

Now substituting these values into the expression:
\[ \frac{y^2 - x^2}{y^2 + 2xy + x^2} = \frac{5621}{5929} \]

### Step 6: Simplify the fraction

Now we check if \( \frac{5621}{5929} \) can be simplified. Since there are no common factors, this fraction is already in its simplest form.

### Final Answer

Thus, the value of \( \frac{y^2 - x^2}{y^2 + 2xy + x^2} \) is:
\[ \frac{5621}{5929} \]