The value of `((0.13)^(2) + (0.21)^(2))/((0.39)^(2) + 81(0.07)^(2)) div ((2.4)^(2) + 3 xx (11.52) + 9)/((2.4)^(6) (2.4)^(4) + 3(17.28))` lies between :

Options

A. 0.4 and 0.5
B. 0.7 and 0.8
C. 0.5 and 0.6
D. 0.6 and 0.7

Answer

To solve the given expression step by step, we will break it down into manageable parts.

### Step 1: Calculate the Numerator of the First Fraction
We start with the expression:
\[ \frac{(0.13)^2 + (0.21)^2}{(0.39)^2 + 81(0.07)^2} \]

First, we calculate \((0.13)^2\) and \((0.21)^2\):
\[ (0.13)^2 = 0.0169 \]
\[ (0.21)^2 = 0.0441 \]

Now, add these two results:
\[ 0.0169 + 0.0441 = 0.0610 \]

Next, we calculate \((0.39)^2\) and \(81(0.07)^2\):
\[ (0.39)^2 = 0.1521 \]
\[ (0.07)^2 = 0.0049 \quad \text{so} \quad 81(0.07)^2 = 81 \times 0.0049 = 0.3969 \]

Now, add these two results:
\[ 0.1521 + 0.3969 = 0.5490 \]

So, the first fraction becomes:
\[ \frac{0.0610}{0.5490} \]

### Step 2: Calculate the Denominator of the First Fraction
Now we calculate the value of the first fraction:
\[ \frac{0.0610}{0.5490} \approx 0.110 \]

### Step 3: Calculate the Numerator of the Second Fraction
Next, we calculate the numerator of the second fraction:
\[ (2.4)^2 + 3 \times 11.52 + 9 \]

Calculate \((2.4)^2\):
\[ (2.4)^2 = 5.76 \]

Now calculate \(3 \times 11.52\):
\[ 3 \times 11.52 = 34.56 \]

Now, add these results:
\[ 5.76 + 34.56 + 9 = 49.32 \]

### Step 4: Calculate the Denominator of the Second Fraction
Now we calculate the denominator of the second fraction:
\[ (2.4)^6 \times (2.4)^4 + 3 \times 17.28 \]

First, calculate \((2.4)^6\) and \((2.4)^4\):
\[ (2.4)^6 = 2.4^2 \times (2.4^4) = 5.76 \times (2.4^4) \quad \text{(we will calculate } (2.4)^4 \text{ next)} \]
\[ (2.4)^4 = (2.4^2)^2 = 5.76^2 = 33.1776 \]

Now calculate:
\[ (2.4)^6 = 5.76 \times 33.1776 \approx 191.102976 \]

Now calculate \(3 \times 17.28\):
\[ 3 \times 17.28 = 51.84 \]

Now add these results:
\[ 191.102976 + 51.84 \approx 242.942976 \]

### Step 5: Calculate the Second Fraction
Now we can find the second fraction:
\[ \frac{49.32}{242.942976} \approx 0.203 \]

### Step 6: Combine the Two Fractions
Now we combine the two fractions:
\[ \frac{0.110}{0.203} \]

Calculating this gives:
\[ \frac{0.110}{0.203} \approx 0.542 \]

### Step 7: Final Result
The value lies between:
\[ 0.5 \text{ and } 0.6 \]