Question
Determine which two values the following irrational number is between.
√15
A.
is between 10 and 11
B.
is between 9 and 10
C.
is between 8 and 9
D.
is between 11 and 12
√15
A.
is between 10 and 11
B.
is between 9 and 10
C.
is between 8 and 9
D.
is between 11 and 12
Answers
GPT-4o mini
To determine the approximate value of \(\sqrt{15}\), we can find the squares of numbers that are close to \( \sqrt{15} \).
First, we know that:
- \(3^2 = 9\)
- \(4^2 = 16\)
Since \(9 < 15 < 16\), it follows that:
\[
3 < \sqrt{15} < 4
\]
Next, we'll narrow this down further by checking the squares of numbers between 3 and 4.
Calculating \(3.8^2\):
\[
3.8^2 = 14.44
\]
Calculating \(3.9^2\):
\[
3.9^2 = 15.21
\]
So we have:
\[
3.8^2 = 14.44 < 15 < 15.21 = 3.9^2
\]
This shows that:
\[
3.8 < \sqrt{15} < 3.9
\]
Next, we can conclude that \( \sqrt{15} \) is between the values \(3.8\) and \(3.9\), which confirms that it is between the whole numbers \(3\) and \(4\).
However, let's compare this result to the options provided. Although we have established it's certainly not between \(9\) and \(10\) or \(10\) and \(11\), we should check the bounds provided in the question more closely.
The key observation here is that \(\sqrt{15} \approx 3.87\). Therefore, the correct answer set does not match closely with the range specified in the options (since none directly show \(3 < \sqrt{15} < 4\)), indicating that the triangle of options allows for the interpretation of values beyond.
With the restrictions set, \(\sqrt{15}\) is definitely not between:
- \(9\) and \(10\)
- \(10\) and \(11\)
- \(11\) and \(12\)
Thus, while not explicitly on the list, the only acceptable assertion matching its inherent mathematical identity, \(\sqrt{15}\) is better compared between:
- \(3\) and \(4\) which suits algebraic formulation.
Thus no valid pairing among provided options exists directly.
Final conclusion: None of the options provided are a match for the actual determination of value nil would apply inherently from \(\sqrt{15}\) resulting hence echoing between numerical roots \(3 < \sqrt{15} < 4\).
First, we know that:
- \(3^2 = 9\)
- \(4^2 = 16\)
Since \(9 < 15 < 16\), it follows that:
\[
3 < \sqrt{15} < 4
\]
Next, we'll narrow this down further by checking the squares of numbers between 3 and 4.
Calculating \(3.8^2\):
\[
3.8^2 = 14.44
\]
Calculating \(3.9^2\):
\[
3.9^2 = 15.21
\]
So we have:
\[
3.8^2 = 14.44 < 15 < 15.21 = 3.9^2
\]
This shows that:
\[
3.8 < \sqrt{15} < 3.9
\]
Next, we can conclude that \( \sqrt{15} \) is between the values \(3.8\) and \(3.9\), which confirms that it is between the whole numbers \(3\) and \(4\).
However, let's compare this result to the options provided. Although we have established it's certainly not between \(9\) and \(10\) or \(10\) and \(11\), we should check the bounds provided in the question more closely.
The key observation here is that \(\sqrt{15} \approx 3.87\). Therefore, the correct answer set does not match closely with the range specified in the options (since none directly show \(3 < \sqrt{15} < 4\)), indicating that the triangle of options allows for the interpretation of values beyond.
With the restrictions set, \(\sqrt{15}\) is definitely not between:
- \(9\) and \(10\)
- \(10\) and \(11\)
- \(11\) and \(12\)
Thus, while not explicitly on the list, the only acceptable assertion matching its inherent mathematical identity, \(\sqrt{15}\) is better compared between:
- \(3\) and \(4\) which suits algebraic formulation.
Thus no valid pairing among provided options exists directly.
Final conclusion: None of the options provided are a match for the actual determination of value nil would apply inherently from \(\sqrt{15}\) resulting hence echoing between numerical roots \(3 < \sqrt{15} < 4\).