jmc

Since $|6+ti|=\sqrt{6^2+t^2}=\sqrt{t^2+36}$, the equation $|6+ti|=10$ tells us that $\sqrt{t^2+36}=10$. Squaring both sides gives $t^2+36=100$, so $t^2=64$. Since we want the positive value of $t$, we have $t=\boxed{8}$.