Question
Nolan used the following procedure to find an estimate for StartRoot 18 EndRoot.
Step 1: Since 4 squared = 16 and 5 squared = 25 and 16 < 18 < 25, StartRoot 18 EndRoot is between 4 and 5.
Step 2: Since 18 is closer to 16, square the tenths closer to 4.
4.1 squared = 16.81
4.2 squared = 17.64
4.3 squared = 18.49
4.4 squared = 19.36
Step 3: Since 18.49 rounds to 18, 4.3 is the best approximation for StartRoot 18 EndRoot.
In which step, if any, did Nolan make an error?
In step 1, StartRoot 18 EndRoot is between 4 and 5 becauseStartRoot 18 EndRoot almost-equals 20 and 4 times 5 = 20.
In step 2, he made a calculation error when squaring.
In step 3, he should have determined which square is closest to 18.
Nolan did not make an error.
Step 1: Since 4 squared = 16 and 5 squared = 25 and 16 < 18 < 25, StartRoot 18 EndRoot is between 4 and 5.
Step 2: Since 18 is closer to 16, square the tenths closer to 4.
4.1 squared = 16.81
4.2 squared = 17.64
4.3 squared = 18.49
4.4 squared = 19.36
Step 3: Since 18.49 rounds to 18, 4.3 is the best approximation for StartRoot 18 EndRoot.
In which step, if any, did Nolan make an error?
In step 1, StartRoot 18 EndRoot is between 4 and 5 becauseStartRoot 18 EndRoot almost-equals 20 and 4 times 5 = 20.
In step 2, he made a calculation error when squaring.
In step 3, he should have determined which square is closest to 18.
Nolan did not make an error.
Answers
GPT-4o mini
Let's analyze each step to determine if Nolan made any errors:
**Step 1**: He correctly identifies that \(4^2 = 16\) and \(5^2 = 25\), which means that \( \sqrt{18} \) falls between 4 and 5. This step is accurate.
**Step 2**: He calculates the squares of the decimal approximations:
- \(4.1^2 = 16.81\)
- \(4.2^2 = 17.64\)
- \(4.3^2 = 18.49\)
- \(4.4^2 = 19.36\)
All of these calculations are correct. So, there is no error in this step.
**Step 3**: Nolan says 4.3 is the best approximation for \( \sqrt{18} \). The value 4.3 squared is 18.49, which is indeed the closest square to 18 according to his calculations, but it is actually slightly greater than 18. To clarify, it should be noted that he should consider which value's square is closest to 18 when deciding; he could also consider the closeness of the squares of 4.2, which is 17.64, to 18.
To summarize:
- **Step 1**: No error.
- **Step 2**: No error in calculations.
- **Step 3**: While 4.3 is a reasonable approximation, 4.2 (squaring to 17.64) is closer to 18 than 4.3 (squaring to 18.49).
Given these points, the most appropriate answer is:
**In step 3, he should have determined which square is closest to 18.** While 4.3 is nearby, 4.2 gives a square that is closer to 18. Therefore, he actually did make an error in his final conclusion.
**Step 1**: He correctly identifies that \(4^2 = 16\) and \(5^2 = 25\), which means that \( \sqrt{18} \) falls between 4 and 5. This step is accurate.
**Step 2**: He calculates the squares of the decimal approximations:
- \(4.1^2 = 16.81\)
- \(4.2^2 = 17.64\)
- \(4.3^2 = 18.49\)
- \(4.4^2 = 19.36\)
All of these calculations are correct. So, there is no error in this step.
**Step 3**: Nolan says 4.3 is the best approximation for \( \sqrt{18} \). The value 4.3 squared is 18.49, which is indeed the closest square to 18 according to his calculations, but it is actually slightly greater than 18. To clarify, it should be noted that he should consider which value's square is closest to 18 when deciding; he could also consider the closeness of the squares of 4.2, which is 17.64, to 18.
To summarize:
- **Step 1**: No error.
- **Step 2**: No error in calculations.
- **Step 3**: While 4.3 is a reasonable approximation, 4.2 (squaring to 17.64) is closer to 18 than 4.3 (squaring to 18.49).
Given these points, the most appropriate answer is:
**In step 3, he should have determined which square is closest to 18.** While 4.3 is nearby, 4.2 gives a square that is closer to 18. Therefore, he actually did make an error in his final conclusion.