If `x=(0.08)^(2), y=(1)/((0.08)^(2)) and z=(1-0.08)^(2)-1` then which of the following relation is true?

Options

A. `y lt z lt x`
B. `z lt x lt y`
C. `y lt x and x=z`
D. `x lt y and x=z`

Answer

To solve the problem, we need to calculate the values of \( x \), \( y \), and \( z \) based on the given definitions and then compare them to find the correct relation.

### Step-by-Step Solution:

1. **Calculate \( x \)**:
\[ x = (0.08)^2 \]
\[ x = 0.08 \times 0.08 = 0.0064 \]

2. **Calculate \( y \)**:
\[ y = \frac{1}{(0.08)^2} \]
\[ y = \frac{1}{0.0064} \]
To calculate \( y \):
\[ y = 156.25 \]

3. **Calculate \( z \)**:
\[ z = (1 - 0.08)^2 - 1 \]
First, calculate \( (1 - 0.08) \):
\[ 1 - 0.08 = 0.92 \]
Now square it:
\[ z = (0.92)^2 - 1 \]
\[ z = 0.8464 - 1 = -0.1536 \]

4. **Compare the values of \( x \), \( y \), and \( z \)**:
- \( x = 0.0064 \)
- \( y = 156.25 \)
- \( z = -0.1536 \)

Now we can arrange them:
\[ z < x < y \]

### Conclusion:
The correct relation is:
\[ z < x < y \]