The minute hand rotates $6^{\circ}$ per minute.  The hour hand rotates $\dfrac{1}{60}\times30^{\circ}=\left (\dfrac{1}{2}\right )^{\circ}$ per mimte.  Starting at $3$ O\textquotesingle clock, the minute hand rotates $\left (6t\right )^{\circ}$ and the hour hand rotates ~$$(90+ \frac{1}{2}t)^{ \circ }$$.  When the minute and hour hands are next perpendicular:  $$6t-\left (90+ \frac{1}{2}t\right )=90$$,  $$\frac{11}{2}t=180$$,  $$t= \frac{360}{11}=32 \frac{8}{11}$$. 
Find the exact number of minutes after $3.00 \text{pm}$ when the minute and hour hands are first at $90^{\circ}$ to each other. 

a: $$31 \frac{2}{11}$$ 
b: $$31 \frac{3}{11}$$ 
c: $$32 \frac{8}{11}$$ 
d: $$33 \frac{3}{11}$$ 
e: None of the above 