As drawn, \(t>1\). Therefore \[\int_1^t\frac2{5x^3}\,dx=\frac19\quad\Longrightarrow\quad\left[-\frac1{5x^2}\right]_1^t=\frac19.\] Thus \[-\frac1{5t^2}+\frac15=\frac19\iff\frac1{5t^2}=\frac4{45}\iff t^2=\frac94.\] Take the positive root consistent with \(t>1\): \(\boxed{t=3/2}\).