To find the whole number and the rational number to the nearest half that \(\sqrt{58}\) falls between, we can first calculate the approximate value of \(\sqrt{58}\).
Calculating \(\sqrt{58}\):
- We know that \(7^2 = 49\) and \(8^2 = 64\).
- Therefore, \(\sqrt{58}\) is between 7 and 8.
For a more precise approximation of \(\sqrt{58}\):
- We can try \(7.5\): \[ 7.5^2 = 56.25 \quad (\text{too low}) \]
- Now try \(7.6\): \[ 7.6^2 = 57.76 \quad (\text{still too low}) \]
- Now try \(7.7\): \[ 7.7^2 = 59.29 \quad (\text{too high}) \]
Since \(7.6^2\) is \(57.76\) (less than \(58\)) and \(7.7^2\) is \(59.29\) (greater than \(58\)), we conclude that: \[ 7.6 < \sqrt{58} < 7.7 \]
Now, rounding to the nearest half, \(7.6\) rounds up to \(7.5\) and \(7.7\) rounds down to \(7.5\) as well. Thus, \(\sqrt{58} \approx 7.62\) falls between the following values:
- Whole Number: The whole number it is between is \(7\) (since it's less than \(8\)).
- Rational Number to the Nearest Half: The rational number to the nearest half it falls between is \(7.5\).
In summary:
- Whole number: 7
- Rational number (to the nearest half): 7.5
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