examday

/TMUA

Triangle $PQR$ has $PQ=5$, $QR=8$, and $\angle PQR=60^\circ$. Point $M$ lies on $QR$ with $QM=3$. An ant starts at $P$ at time $t=0$ and walks in a straight line directly to $M$ at constant speed, arriving at $M$ at time $t=6$. At the same instant $t=0$, a second ant starts at $Q$ and walks directly to $R$ along $QR$ at constant speed, arriving at $R$ at time $t=8$. Find the shortest distance between the two ants at any point during the interval while both ants are still moving (i.e. for $0\le t\le6$).