Question

-) Determine the perfect square values that best complete the expression sqrt(x) < sqrt(300) < sqrt(y) by squeezing the interval on a number line.

(1 point)

x = 256 y = 361

x = 299 y - 301

x = 289 y = 324

x = 100 y = 400

Answer 1

To determine the perfect square values that best complete the expression \( \sqrt{x} < \sqrt{300} < \sqrt{y} \), we need to find perfect squares for \( x \) and \( y \) such that:

1. \( \sqrt{x} < \sqrt{300} \) implies \( x < 300 \)
2. \( \sqrt{300} < \sqrt{y} \) implies \( 300 < y \)

Now, let's calculate \( \sqrt{300} \):

\[
\sqrt{300} \approx 17.32 \quad (\text{since } 17^2 = 289 \text{ and } 18^2 = 324)
\]

Thus, we need perfect squares for \( x \) that are less than 300 and for \( y \) that are greater than 300.

Now, we can analyze the options:

1. **x = 256, y = 361**
- \( \sqrt{256} = 16 < \sqrt{300} \approx 17.32 < \sqrt{361} = 19 \)
- This pair is valid.

2. **x = 299, y = 301**
- \( \sqrt{299} < \sqrt{300} \) is not true since \( \sqrt{299} \approx 17.29 \) and is not less than 17.32.
- This pair is invalid.

3. **x = 289, y = 324**
- \( \sqrt{289} = 17 < \sqrt{300} \) and \( \sqrt{300} < \sqrt{324} = 18 \)
- This pair is valid.

4. **x = 100, y = 400**
- \( \sqrt{100} = 10 < \sqrt{300} \) but \( \sqrt{300} < \sqrt{400} = 20 \), which is also valid.
- However, these values are much farther apart than others.

Evaluating the options, \( \boxed{256} \) for \( x \) and \( \boxed{361} \) for \( y \) (the first option) along with \( \boxed{289} \) for \( x \) and \( \boxed{324} \) for \( y \) (the third option) are both valid. The first option (256, 361) suggests a tighter squeeze around \( \sqrt{300} \).

Thus, the best option that completes the expression while squeezing the interval on a number line is \( x = 256 \) and \( y = 361 \).