Answer
See the answer below.
Work Step by Step
Since $\sqrt{\bar{u^2}}=\sqrt{3RT/M}$, to increase the rms speed by 10% we need to increase $\sqrt{T}$ by 10% so:
$\sqrt{T'}=1.1\sqrt{240}$
$T'=290.4\ K=17.4\ °C$
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