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Four students provide the following approximations for [tex]\sqrt{0.89}[/tex]:

- Anyah states that [tex]\sqrt{0.89}[/tex] is between 0.44 and 0.45 because [tex]0.44 \ \textless \ \frac{0.89}{2} \ \textless \ 0.45[/tex].
- Matthew states that [tex]\sqrt{0.89}[/tex] is between 0 and 1 because [tex]0 \ \textless \ 0.89 \ \textless \ 1[/tex].
- Rhoda states that [tex]\sqrt{0.89}[/tex] is between 0.9 and 1.0 because [tex]0.9^2 \ \textless \ 0.89 \ \textless \ 1.0^2[/tex].
- Ming states that [tex]\sqrt{0.89}[/tex] is between 0.93 and 0.95 because [tex](0.93)^2 \ \textless \ 0.89 \ \textless \ (0.95)^2[/tex].

Which solution(s) are correct?

A. Anyah's only
B. Ming's only
C. Anyah's, Matthew's, and Rhoda's
D. Matthew's, Rhoda's, and Ming's



Answer :

GinnyAnswer
Let's analyze each student's statement step by step to determine the correctness of each approximation for [tex]\(\sqrt{0.89}\)[/tex].

1. Anyah’s Approximation:
- Anyah claims that [tex]\(\sqrt{0.89}\)[/tex] is between 0.44 and 0.45, reasoning that [tex]\(0.44 < \frac{0.89}{2} < 0.45\)[/tex].
- Calculation: [tex]\(\frac{0.89}{2} = 0.445\)[/tex].
- We need to check if [tex]\(0.44 < 0.445 < 0.45\)[/tex].
- Clearly, [tex]\(0.445 < 0.45\)[/tex] is correct.
- However, [tex]\(0.445\)[/tex] is not less than [tex]\(0.44\)[/tex].
- Therefore, Anyah's statement is incorrect.

2. Matthew’s Approximation:
- Matthew states that [tex]\(\sqrt{0.89}\)[/tex] is between 0 and 1, because [tex]\(0 < 0.89 < 1\)[/tex].
- This assertion uses the fact that the square root of any positive number between 0 and 1 will also lie between 0 and 1.
- Since [tex]\(\sqrt{0.89}\)[/tex] is indeed a value between 0 and 1, Matthew’s statement is correct.

3. Rhoda’s Approximation:
- Rhoda claims that [tex]\(\sqrt{0.89}\)[/tex] is between 0.9 and 1.0, using the inequalities [tex]\(0.9^2 < 0.89 < 1.0^2\)[/tex].
- Calculation:
- [tex]\(0.9^2 = 0.81\)[/tex]
- [tex]\(1.0^2 = 1\)[/tex]
- Therefore, we need to check if [tex]\(0.81 < 0.89 < 1\)[/tex], which is true.
- Since the square root of 0.89 should lie between the square roots of 0.81 and 1, and [tex]\(\sqrt{0.81} = 0.9\)[/tex] and [tex]\(\sqrt{1} = 1\)[/tex], Rhoda’s statement is correct.

4. Ming’s Approximation:
- Ming suggests that [tex]\(\sqrt{0.89}\)[/tex] is between 0.93 and 0.95, reasoning that [tex]\((0.93)^2 < 0.89 < (0.95)^2\)[/tex].
- Calculation:
- [tex]\(0.93^2 = 0.8649\)[/tex]
- [tex]\(0.95^2 = 0.9025\)[/tex]
- So, we need to check if [tex]\(0.8649 < 0.89 < 0.9025\)[/tex], which is true.
- Hence, Ming’s statement is also correct.

Based on the analysis provided:

- Anyah’s statement is incorrect.
- Matthew’s statement is correct.
- Rhoda’s statement is correct.
- Ming’s statement is correct.

Therefore, the correct solutions are Matthew's, Rhoda's, and Ming's. The final answer is:

Matthew's, Rhoda's, and Ming's

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