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If 1 rotation=360 degrees is correct, this is my answer.
This, I think, is ratio and proportion.

1 rotation=360 degrees
1 rotation=30 minutes
1 rotation    (cross product)   
30 minutes             12 minutes

=12 minutes revolution / 30 minutes
=6/15 rotation (the minutes is cancelled)
=2/5 rotation
2/5 rotation x 360 degrees
                     1 rotation
= 2/5 x 360 degrees (the rotation is cancelled)
=720/5 degrees
=144 degrees
Therefore, the measure of the angle that a cart will rotate in 12 minutes is 144 degrees.

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