Out of the numbers `0.3`, `0.03`, `0.9`, `0.09` the number that is nearest to the value of `sqrt(0.9)` is

Options

A. A) `0.3`
B. B) `0.03`
C. C) `0.9`
D. D) `0.09`

Answer

To find the number that is nearest to the value of \( \sqrt{0.9} \) from the options \( 0.3, 0.03, 0.9, \) and \( 0.09 \), we can follow these steps:

### Step 1: Calculate \( \sqrt{0.9} \)
We can express \( 0.9 \) as \( \frac{9}{10} \):
\[ \sqrt{0.9} = \sqrt{\frac{9}{10}} = \frac{\sqrt{9}}{\sqrt{10}} = \frac{3}{\sqrt{10}} \]

### Step 2: Rationalize \( \frac{3}{\sqrt{10}} \)
To simplify \( \frac{3}{\sqrt{10}} \), we can multiply the numerator and denominator by \( \sqrt{10} \):
\[ \frac{3}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{3\sqrt{10}}{10} \]

### Step 3: Estimate \( \sqrt{10} \)
We know that \( \sqrt{10} \) is approximately \( 3.16 \). Therefore:
\[ \frac{3\sqrt{10}}{10} \approx \frac{3 \times 3.16}{10} = \frac{9.48}{10} \approx 0.948 \]

### Step 4: Compare \( 0.948 \) with the given options
Now we need to compare \( 0.948 \) with the given options:
- \( 0.3 \)
- \( 0.03 \)
- \( 0.9 \)
- \( 0.09 \)

### Step 5: Determine which option is closest to \( 0.948 \)
- The difference between \( 0.948 \) and \( 0.3 \) is \( |0.948 - 0.3| = 0.648 \)
- The difference between \( 0.948 \) and \( 0.03 \) is \( |0.948 - 0.03| = 0.918 \)
- The difference between \( 0.948 \) and \( 0.9 \) is \( |0.948 - 0.9| = 0.048 \)
- The difference between \( 0.948 \) and \( 0.09 \) is \( |0.948 - 0.09| = 0.858 \)

### Step 6: Identify the nearest number
The smallest difference is \( 0.048 \) which corresponds to \( 0.9 \). Therefore, the number that is nearest to \( \sqrt{0.9} \) is:
\[ \boxed{0.9} \]