To find where \( \sqrt{58} \) lies on the number line, we first need to approximate the value of \( \sqrt{58} \).
Calculating some nearby perfect squares:
- \( 7^2 = 49 \)
- \( 8^2 = 64 \)
Since \( 49 < 58 < 64 \), it follows that:
\[ 7 < \sqrt{58} < 8 \]
To narrow it down further, let's calculate \( 7.5^2 \):
\[ 7.5^2 = 56.25 \]
Now we check \( 7.5 \) in our range:
\[ 56.25 < 58 \]
This tells us \( \sqrt{58} > 7.5 \).
Next, we can check \( 7.6 \):
\[ 7.6^2 = 57.76 \]
Thus:
\[ 57.76 < 58 \]
This confirms \( \sqrt{58} > 7.6 \).
Now, checking \( 7.7 \):
\[ 7.7^2 = 5929 \]
So:
\[ 59.29 > 58 \]
This confirms \( \sqrt{58} < 7.7 \).
So far we have:
\[ 7.6 < \sqrt{58} < 7.7 \]
To characterize \( \sqrt{58} \):
The range \( 7.5 < \sqrt{58} < 7.7 \) also suggests that \( \sqrt{58} \) might be explored further with the next half point being \( 7.65 \):
Calculating \( 7.65^2 \):
\[ 7.65^2 = 58.5225 \]
Since \( 58.5225 > 58 \), we can tell \( 7.65 > \sqrt{58} \).
Thus, we can conclude:
\[ 7.6 < \sqrt{58} < 7.65 \]
This means \( \sqrt{58} \) is indeed between \( 7.5 \) and \( 8 \), and therefore the closest rational number to the nearest half would suggest \( \sqrt{58} \) fits neatly into the given intervals.
Thus, the answer is:
D. between 7.5 and 8.
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