What is the square root of 99999?

$316^2 = 99856$ and $317^2 = 100489$

$99999$ lies between them, so $\sqrt{99999}$ is between $316$ and $317$.

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Quick estimate: $\sqrt{N} \approx a + \dfrac{N - a^2}{2a}$, with $a = 316$:

$\sqrt{99999} \approx 316 + \dfrac{99999 - 99856}{2 \times 316}$

$= 316 + \dfrac{143}{632}$

$\approx 316.226$

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The options differ only in the second decimal, so compare with the midpoint $316.225$:

$316.225^2 = 316^2 + 2 \times 316 \times 0.225 + 0.225^2$

$= 99856 + 142.2 + 0.050625$

$= 99998.250625$

$99999$ is greater than this, so $\sqrt{99999}$ is greater than $316.225$.

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$\sqrt{99999} = 316.226\ldots$

Rounded to two decimal places, $\sqrt{99999} = 316.23$.