To find the whole number and rational number to the nearest half that \(\sqrt{58}\) lies between, we first need to estimate \(\sqrt{58}\).
We know that:
- \(\sqrt{49} = 7\)
- \(\sqrt{64} = 8\)
Since \(49 < 58 < 64\), it follows that: \[ 7 < \sqrt{58} < 8 \]
Next, we can find a more precise approximation of \(\sqrt{58}\) by trying some values:
- Calculating \(7.5^2 = 56.25\)
- Calculating \(7.6^2 = 57.76\)
- Calculating \(7.7^2 = 59.29\)
From this, we see that: \[ 7.5^2 < 58 < 7.6^2 \] This means: \[ 7.5 < \sqrt{58} < 7.6 \]
Now let's round to the nearest half. The nearest half around \(\sqrt{58}\) is \(7.5\).
So, to summarize:
- The whole number that \(\sqrt{58}\) is between is \(7\) and \(8\).
- The nearest half rational number that \(\sqrt{58}\) falls between (and is also a boundary) is \(7.5\).
Thus, \(\sqrt{58}\) is between the whole number \(7\) and the rational number \(7.5\).