jmc

Since the candle is $119$ centimeters tall, the time the candle takes to burn down is $$T=\sum _{k=1}^{119}10k=10\cdot \frac{119\cdot 120}{2}=71400.$$We want to compute the height of the candle at time $\frac{T}{2}=35700$ seconds. Suppose that, at this time, the first $m$ centimeters have burnt down completely, but not the $(m+1)$st centimeter completely. Then we must have $$\sum _{k=1}^{m}10k\le 35700<\sum _{k=1}^{m+1}10k$$(the first quantity is the time it takes for the first $m$ centimeters to burn down; the last is the time it takes for the first $(m+1)$ centimeters to burn down). This simplifies to $$5m(m+1)\le 35700<5(m+1)(m+2).$$To find $m$, we note that we should have $5m^2\approx 35700$, or $m^2\approx 7140$, so $m\approx 85$. Trying values of $m$, we find that when $m=84$, $$5m(m+1)=35700$$exactly. Therefore, at time $\frac{T}{2}$, exactly the first $84$ centimeters have burnt down, and nothing more, so the height of the remaining part of the candle is $119-84=\boxed{35}$ centimeters.