Answer
The exit velocity of the lava is $348.704$ feet/second or $237.753$ miles/hour.
Work Step by Step
$$s=v_0t-16t^2(ft)$$
Find the velocity function of the lava, $v$: $$v=\frac{ds}{dt}=v_0-32t(ft/sec)$$
The lava's exit velocity is the highest, then as the lava reaches higher, its velocity diminishes. So as the lava reaches its peak point, the velocity $v=0$. In other words, $$v_0-32t=0$$ $$v_0=32t$$
Using $v_0=32t$ to substitute the formula for $s$: $$s=32t\times t-16t^2=32t^2-16t^2=16t^2$$
Now take $s=1900ft$, we can find the time $t$ it takes for the lava to reach that height: $$16t^2=1900$$ $$t^2=118.75$$ $$t\approx10.897(sec)$$
Eventually, we can find the exit velocity $v_0$: $$v_0=32t\approx348.704ft/sec\approx237.753mi/hr$$
