Note that $B$ is the intersection of the line $y=\frac{1}{2}x+5$ and $x^2+y^2=100$ in the first quadrant, which means that the coordinates of $B$ must both be positive. We find the coordinates of $B$ to be $(6,8)$. The perpendicular slope of $l_1$ is $-2$. So the equation of the perpendicular line through $B$ is $y-8=-2(x-6)$. We seek the $y$-intercept, so $\boxed{t=20}$