Answer
$t=\frac{15}{32} \text{sec}$
Work Step by Step
RECALL:
$\omega=\dfrac{\theta}{t}$
Substitute the given values into the given formula above to obtain:
$\omega=\dfrac{\theta}{t}
\\\frac{8\pi}{9} \text{ radians per sec}= \dfrac{\frac{5\pi}{12}\text{ radians}}{t}$
Cross-multiply to obtain:
\begin{array}{ccc}
&t \cdot (\frac{8\pi}{9}\text{ radians per sec})&=&\frac{5\pi}{12} \text{radians}
\\&\dfrac{t \cdot (\frac{8\pi}{9}\text{ radians per sec})}{\frac{8\pi}{9}\text{ radians per sec}}&=&\dfrac{\frac{5\pi}{12} \text{radians}}{\frac{8\pi}{9}\text{ radians per sec}}
\\&t &= &\frac{15}{32} \text{sec}\end{array}
