Parametric, Polar and Vector-Valued Functions · Hard problem

A bead slides along a wire shaped like $\mathbf{r}(t)=\left\langle t^2,\;\dfrac{2}{3}t^3\right\rangle$ ($t\ge0$, length in cm). Find the time $t$ at which the bead has traveled a total arc length of $\dfrac{52}{3}$ cm from $t=0$.