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Nolan used the following procedure to find an estimate for [tex]\sqrt{18}[/tex].

Step 1: Since [tex]4^2 = 16[/tex] and [tex]5^2 = 25[/tex], and [tex]16 \ \textless \ 18 \ \textless \ 25[/tex], [tex]\sqrt{18}[/tex] is between 4 and 5.

Step 2: Since 18 is closer to 16, square the tenths closer to 4.
[tex]\[
\begin{array}{l}
4.1^2 = 16.81 \\
4.2^2 = 17.64 \\
4.3^2 = 18.49 \\
4.4^2 = 19.36
\end{array}
\][/tex]

Step 3: Since 18.49 rounds to 18, 4.3 is the best approximation for [tex]\sqrt{18}[/tex].

In which step, if any, did Nolan make an error?

A. In Step 1, [tex]\sqrt{18}[/tex] is between 4 and 5 because [tex]\sqrt{18} = 20[/tex] and [tex]4 \times 5 = 20[/tex].

B. In Step 2, he made a calculation error when squaring.

C. In Step 3, he should have determined which square is closest to 18.

D. Nolan did not make an error.



Answer :

GinnyAnswer
To determine whether Nolan made an error in his steps, we will analyze each step in detail based on the provided results.

Step 1: Finding the interval for [tex]\(\sqrt{18}\)[/tex]

Nolan claims that [tex]\(\sqrt{18}\)[/tex] is between 4 and 5 because [tex]\(4^2 = 16\)[/tex] and [tex]\(5^2 = 25\)[/tex], and [tex]\(16 < 18 < 25\)[/tex].

From the result:
- [tex]\(\sqrt{18} \approx 4.2426\)[/tex]

Since [tex]\(4 < 4.2426 < 5\)[/tex], step 1 is correct. [tex]\(\sqrt{18}\)[/tex] is indeed between 4 and 5.

Step 2: Squaring numbers close to 4

Nolan provides the calculations:

[tex]\[ \begin{align*} 4.1^2 & = 16.81 \\ 4.2^2 & = 17.64 \\ 4.3^2 & = 18.49 \\ 4.4^2 & = 19.36 \end{align*} \][/tex]

Comparing these values with the results:
- [tex]\(4.1^2 = 16.81\)[/tex]
- [tex]\(4.2^2 = 17.64\)[/tex]
- [tex]\(4.3^2 = 18.49\)[/tex]
- [tex]\(4.4^2 = 19.36\)[/tex]

All values match those provided, so there is no calculation error. Nolan did not make an error in step 2.

Step 3: Determining the best approximation

Nolan concludes that since [tex]\(18.49\)[/tex] rounds to 18, [tex]\(4.3\)[/tex] is the best approximation for [tex]\(\sqrt{18}\)[/tex].

Given the calculations from step 2:
- [tex]\(4.3^2 = 18.49\)[/tex]
- [tex]\(4.4^2 = 19.36\)[/tex]

Since [tex]\(18.49\)[/tex] is closer to 18 than [tex]\(19.36\)[/tex],
[tex]\[ \left|18.49 - 18\right| < \left|19.36 - 18\right| \][/tex]

Thus, [tex]\(4.3\)[/tex] is indeed the closest approximation to 18. Nolan correctly identified [tex]\(4.3\)[/tex] as the best approximation.

Conclusion:

After carefully reviewing each step:
1. [tex]\(\sqrt{18} \approx 4.2426\)[/tex] confirms [tex]\(\sqrt{18}\)[/tex] is between 4 and 5.
2. His squaring calculations were correct.
3. [tex]\(4.3\)[/tex] is closer to 18 than [tex]\(4.4\)[/tex].

Nolan did not make an error in any of his steps.

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