Let’s look at the steps Nolan took and verify their correctness.
### Step 1
Nolan correctly identified that since [tex]\(4^2 = 16\)[/tex] and [tex]\(5^2 = 25\)[/tex], and because [tex]\(16 < 18 < 25\)[/tex], [tex]\(\sqrt{18}\)[/tex] must lie between 4 and 5. This step is accurate.
### Step 2
Nolan performed calculations to square numbers that are in the range from 4.1 to 4.4:
[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]
These calculations are correct.
### Step 3
In step 3, Nolan concluded that [tex]\(4.3^2 = 18.49\)[/tex] rounds to 18, making 4.3 the best approximation for [tex]\(\sqrt{18}\)[/tex]. However, this step needs closer inspection to check for accuracy in approximation:
The actual result of the calculations are:
[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]
By evaluating these results:
- [tex]\(16.81\)[/tex] is less than 18.
- [tex]\(17.64\)[/tex] is less than 18.
- [tex]\(18.49\)[/tex] is greater than 18.
- [tex]\(19.36\)[/tex] is greater than 18.
Since [tex]\(18.49\)[/tex] is greater than 18, we note that [tex]\(4.2^2 = 17.64\)[/tex] is actually closer to 18 than [tex]\(4.3^2 = 18.49\)[/tex]. Thus, the error is in determining the closest value. Nolan should have identified that [tex]\(17.64\)[/tex] (from [tex]\(4.2^2\)[/tex]) is closer to 18 than [tex]\(18.49\)[/tex] (from [tex]\(4.3^2\)[/tex]).
Therefore, the error occurred in Step 3, where Nolan should have determined that [tex]\( \sqrt{18} \)[/tex] is best approximated by 4.2, not 4.3.
