Question

1. as shown, the number line between 0 and 8 is divided into 6 segments of equal length by points a through e. which of the following statements about \\(\sqrt{8}\\) is true?

\\(\longleftarrow 0\quad a\quad b\quad c\quad d\quad e\quad 8\longrightarrow\\)
f. \\(\sqrt{8}\\) is at c.
g. \\(\sqrt{8}\\) is at d.
h. \\(\sqrt{8}\\) is between a and b.
j. \\(\sqrt{8}\\) is between b and c.

Explanation:

Step1: Find length of each segment

The total length from 0 to 8 is \( 8 - 0 = 8 \). There are 6 segments, so the length of each segment is \( \frac{8}{6}=\frac{4}{3}\approx1.333 \).

Step2: Find the approximate value of \( \sqrt{8} \)

Simplify \( \sqrt{8} \): \( \sqrt{8} = 2\sqrt{2}\approx2\times1.414 = 2.828 \).

Step3: Determine the positions of points A, B, C, D, E

• Point A: \( 0+\frac{4}{3}\approx1.333 \)
• Point B: \( 1.333+\frac{4}{3}\approx1.333 + 1.333 = 2.666 \)
• Point C: \( 2.666+\frac{4}{3}\approx2.666+1.333 = 3.999\approx4 \)
• Point D: \( 4+\frac{4}{3}\approx5.333 \)
• Point E: \( 5.333+\frac{4}{3}\approx6.666 \)

Step4: Compare \( \sqrt{8}\approx2.828 \) with the positions

We see that \( 2.666\lt2.828\lt3.999 \), which means \( \sqrt{8} \) is between B and C.

Answer:

J. \( \sqrt{8} \) is between B and C.