Using rational approximations, what statement is true?

Question

A. √48 ≤ 7
B. √ 49 ≥ 7
C. √ 48 ≤ √36
D. √ 48 ≥ √ 36

Answer 1

To evaluate the statements using rational approximations, we first need to calculate the values of the square roots involved.

1. Calculate \( \sqrt{48} \): \[ \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \approx 4 \times 1.732 = 6.928 \text{ (approx)} \]

2. Calculate \( \sqrt{49} \): \[ \sqrt{49} = 7 \]

3. Calculate \( \sqrt{36} \): \[ \sqrt{36} = 6 \]

Now, we will evaluate each of the statements:

A. \( \sqrt{48} \leq 7 \)
From our approximation, \( 6.928 \leq 7 \) is true.

B. \( \sqrt{49} \geq 7 \)
Since \( \sqrt{49} = 7 \), it is also true that \( 7 \geq 7 \).

C. \( \sqrt{48} \leq \sqrt{36} \)
From our calculations, \( 6.928 \leq 6 \) is false.

D. \( \sqrt{48} \geq \sqrt{36} \)
Since \( 6.928 \geq 6 \) is true.

Summary of Findings:

• Statement A is true.
• Statement B is true.
• Statement C is false.
• Statement D is true.

Thus, the true statements are A, B, and D.