3. The room temperature $T$ (unit: Celsius) as a function of time $t$ (unit: hours) is: $T=$ $a \sin t+b \cos t, t \in(0,+\infty)$, where $a, b$ are positive real numbers. If the maximum temperature difference in the room is 10 degrees Celsius, then the maximum value of $a+b$ is $\qquad$
$T=\sqrt{a^{2}+b^{2}} \sin (t+\varphi)$, where $\varphi \in\left(0, \frac{\pi}{2}\right)$ and $\tan \varphi=\frac{b}{a}$. Thus, $2 \sqrt{a^{2}+b^{2}}=10 \Rightarrow a^{2}+b^{2}=25 \Rightarrow(a+b)^{2} \leqslant 2\left(a^{2}+b^{2}\right)=50 \Rightarrow a+b \leqslant 5 \sqrt{2}$, equality holds when $a=b=\frac{5}{2} \sqrt{2}$. Therefore, the maximum value of $a+b$ is $5 \sqrt{2}$.