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. 
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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, when the minute and hour hands are next perpendicular:  $$(90+90)\div(6-0.5)=\frac{360}{11}=32 \frac{8}{11}$$ 