#### Answer

To measure $1L$ of water within the required error, we need to measure $h$ within the range of $[87.535mm,89.304mm]$.

#### Work Step by Step

$$V=36\pi h$$
To find how closely we must measure $h$ to measure $1000cm^3$ of water with an error of no more than $10cm^3$, basically is to find an interval $[a, b]$ of $h$ values such that for all $h\in[a,b]$: $$|36\pi h-1000|\le10$$
$$-10\le36\pi h-1000\le10$$
$$990\le36\pi h\le1010$$
$$\frac{990}{36\pi}\le h\le\frac{1010}{36\pi}$$
$$8.7535cm\le h\le8.9304cm$$
$$87.535mm\le h\le89.304mm$$
Therefore, to measure $1L$ of water within the required error, we need to measure $h$ within the range of $[87.535mm,89.304mm]$.