Understanding the Square Root Problem

The question asks us to calculate the value of the expression: \(\sqrt{\frac{36.1}{102.4}}\). This involves finding the square root of a fraction that contains decimal numbers.

Simplifying the Fraction

To make the calculation easier, we can first remove the decimal points. We can do this by multiplying both the numerator and the denominator inside the square root by 10: \(\sqrt{\frac{36.1 \times 10}{102.4 \times 10}} = \sqrt{\frac{361}{1024}}\).

Now, the problem is to find the square root of the fraction \(\frac{361}{1024}\). We can find the square root of the numerator and the denominator separately.

Calculating the Square Roots

We need to find two values:

• The square root of the numerator, 361.
• The square root of the denominator, 1024.

Let's find the square root of 361. We can test numbers. We know that \(10 \times 10 = 100\) and \(20 \times 20 = 400\). So the square root must be between 10 and 20. Let's try numbers ending in 1 or 9. \(19 \times 19 = 361\). So, \(\sqrt{361} = 19\).

Now let's find the square root of 1024. We know that \(30 \times 30 = 900\) and \(40 \times 40 = 1600\). So the square root must be between 30 and 40. Since 1024 ends in 4, the square root must end in 2 or 8. Let's try 32. \(32 \times 32 = 1024\). So, \(\sqrt{1024} = 32\).

Determining the Final Answer

Now we can combine the results:

\(\sqrt{\frac{361}{1024}} = \frac{\sqrt{361}}{\sqrt{1024}} = \frac{19}{32}\).

Comparing this result with the given options, we find that it matches the third option.