By assuming `sqrt(13)=3.605` (approximate) `sqrt(130)=11.40` (approximate), find the value of `sqrt(1.3)+sqrt(1300)+sqrt(0.013)`

Options

A. 36.164
B. 37.304
C. 36.304
D. 37.164

Answer

To find the value of \( \sqrt{1.3} + \sqrt{1300} + \sqrt{0.013} \), we can use the approximations provided for \( \sqrt{13} \) and \( \sqrt{130} \).

### Step 1: Calculate \( \sqrt{1.3} \)

We can express \( \sqrt{1.3} \) in terms of \( \sqrt{13} \):

\[ \sqrt{1.3} = \sqrt{\frac{13}{10}} = \frac{\sqrt{13}}{\sqrt{10}} \]

Given that \( \sqrt{13} \approx 3.605 \) and \( \sqrt{10} \approx 3.162 \) (since \( \sqrt{10} \) can be approximated as \( \sqrt{100/10} = 10/3.162 \approx 3.162 \)), we can calculate:

\[ \sqrt{1.3} \approx \frac{3.605}{3.162} \approx 1.141 \]

### Step 2: Calculate \( \sqrt{1300} \)

Next, we express \( \sqrt{1300} \):

\[ \sqrt{1300} = \sqrt{13 \times 100} = \sqrt{13} \times \sqrt{100} = \sqrt{13} \times 10 \]

Using the approximation for \( \sqrt{13} \):

\[ \sqrt{1300} \approx 3.605 \times 10 = 36.05 \]

### Step 3: Calculate \( \sqrt{0.013} \)

Now, we can express \( \sqrt{0.013} \):

\[ \sqrt{0.013} = \sqrt{\frac{13}{1000}} = \frac{\sqrt{13}}{\sqrt{1000}} = \frac{\sqrt{13}}{10 \sqrt{10}} \]

Using the approximations:

\[ \sqrt{0.013} \approx \frac{3.605}{10 \times 31.62} \approx \frac{3.605}{316.2} \approx 0.0114 \]

### Step 4: Combine the results

Now we can combine all the approximated values:

\[ \sqrt{1.3} + \sqrt{1300} + \sqrt{0.013} \approx 1.141 + 36.05 + 0.0114 \]

Calculating this gives:

\[ 1.141 + 36.05 + 0.0114 \approx 37.2024 \]

### Final Result

Thus, the value of \( \sqrt{1.3} + \sqrt{1300} + \sqrt{0.013} \) is approximately:

\[ \boxed{37.2024} \]