`(1)/(4)xx sqrt``([(12.1)^(2)-(8.1)^(2)]` `div``[(0.25)^(2)+(0.25)(19.95)])`.

Options

A. 8
B. 4
C. 1
D. 3

Answer

To solve the expression \(\frac{1}{4} \times \sqrt{(12.1)^2 - (8.1)^2} \div [(0.25)^2 + (0.25)(19.95)]\), we will follow these steps:

### Step 1: Simplify the expression inside the square root using the difference of squares formula.
The difference of squares formula states that \(a^2 - b^2 = (a - b)(a + b)\). Here, let \(a = 12.1\) and \(b = 8.1\).

\[ (12.1)^2 - (8.1)^2 = (12.1 - 8.1)(12.1 + 8.1) \]

Calculating \(12.1 - 8.1\) and \(12.1 + 8.1\):

\[ 12.1 - 8.1 = 4 \quad \text{and} \quad 12.1 + 8.1 = 20.2 \]

So,

\[ (12.1)^2 - (8.1)^2 = 4 \times 20.2 \]

### Step 2: Substitute back into the expression.
Now we substitute back into the original expression:

\[ \frac{1}{4} \times \sqrt{4 \times 20.2} \div [(0.25)^2 + (0.25)(19.95)] \]

### Step 3: Simplify the denominator.
Calculate \((0.25)^2\) and \((0.25)(19.95)\):

\[ (0.25)^2 = 0.0625 \]
\[ (0.25)(19.95) = 4.9875 \]

Now add these two results:

\[ 0.0625 + 4.9875 = 5.05 \]

### Step 4: Rewrite the expression.
Now we have:

\[ \frac{1}{4} \times \sqrt{4 \times 20.2} \div 5.05 \]

### Step 5: Simplify the square root.
Calculate \(\sqrt{4 \times 20.2}\):

\[ \sqrt{4 \times 20.2} = \sqrt{80.8} = 4 \sqrt{5.05} \]

### Step 6: Substitute back into the expression.
Now the expression becomes:

\[ \frac{1}{4} \times 4 \sqrt{5.05} \div 5.05 \]

### Step 7: Cancel the 4's.
The 4's cancel out:

\[ \sqrt{5.05} \div 5.05 \]

### Step 8: Convert division to multiplication.
This can be rewritten as:

\[ \frac{\sqrt{5.05}}{5.05} = \frac{1}{\sqrt{5.05}} \]

### Step 9: Rationalize the denominator.
To rationalize, multiply numerator and denominator by \(\sqrt{5.05}\):

\[ \frac{\sqrt{5.05}}{5.05} = \frac{1}{\sqrt{5.05}} \times \frac{\sqrt{5.05}}{\sqrt{5.05}} = \frac{\sqrt{5.05}}{5.05} \]

This simplifies to \(1\).

### Final Answer:
Thus, the final answer is:

\[ \boxed{1} \]