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Kaito is making a square sign. He wants to be sure that each side of the sign measures as a rational number. Which of the following values could be the area of Kaito's sign, in inches?
39.65
42.25
43.71
45.15

Answer 1

To determine which of the given values could be the area of Kaito's square sign, we recall that the area \(A\) of a square is given by the formula:

\[
A = s^2
\]

where \(s\) is the length of one side of the square. For \(s\) to be a rational number, \(A\) must be a perfect square of a rational number.

To check if a number can be the area of a square with rational sides, we can look for rational square roots. This means we should see if the square root of each of the options provided is a rational number.

Let’s evaluate the square roots of the given areas:

1. **\(39.65\)**:
\[
\sqrt{39.65} \approx 6.293
\]
(Not a rational number)

2. **\(42.25\)**:
\[
\sqrt{42.25} = 6.5
\]
(Rational number)

3. **\(43.71\)**:
\[
\sqrt{43.71} \approx 6.6
\]
(Not a rational number)

4. **\(45.15\)**:
\[
\sqrt{45.15} \approx 6.71
\]
(Not a rational number)

Based on this analysis, the only value that could be the area of Kaito's sign with rational side lengths is:

**42.25**.