The squared distance between two points on the surface of a spherical balloon is defined by $,d{}_2^2=d{}^{2}si{n}^{2}d{}^2.$ Calculate the distance between two points that lie on the plane $=\frac{}{2}$ and whose azimuthal coordinates are $=\frac{}{2}$ and $=.$ If the radius of an expanding spherical balloon at any given time is given by $R(t)=\sqrt[\phantom{2}]{\frac{t}{t_0}},$ then at what time is the distance between the two given points double that at $t_0?$