Question

If √x
has a decimal approximation of 3.4, then x
is between which two integers? In your final answer, include all calculations.


WRITER

Answer 1

To find the integer values that \( x \) is between when \( \sqrt{x} \) has a decimal approximation of 3.4, we start by squaring both sides of \( \sqrt{x} \approx 3.4 \) to express \( x \).

\[
\sqrt{x} \approx 3.4
\]

Squaring both sides gives:

\[
x \approx (3.4)^2
\]

Calculating \( (3.4)^2 \):

\[
3.4 \times 3.4 = 11.56
\]

This means \( x \) is approximately 11.56. Now we need to determine the integers between which this value lies.

The integers between which \( 11.56 \) falls are 11 and 12.

To confirm, we can check the square roots of both 11 and 12:

1. **Calculating \( \sqrt{11} \)**:

\[
\sqrt{11} \approx 3.3166 \quad \text{(which is less than 3.4)}
\]

2. **Calculating \( \sqrt{12} \)**:

\[
\sqrt{12} \approx 3.4641 \quad \text{(which is greater than 3.4)}
\]

Since \( \sqrt{11} < 3.4 < \sqrt{12} \), we can conclude that:

\[
\sqrt{x} \approx 3.4 \Rightarrow x \text{ is between } 11 \text{ and } 12.
\]

Thus, the final answer is that \( x \) is between **11 and 12**.