A1000-kg barge is being towed by means of two horizontal cables. one cable is pulling with a force of 80.0 n in a direction 30.0° west of north. the second cable pulls in a direction 20.0° east of north. what should the magnitude of its pulling force be so that the barge will accelerate northward?

To obtain an acceleration northward is necessary than the west component of the first cable and the east component of the second cable be equal, otherwise the barge will head northwest or northeast.  
So for the first cable the component to west is:
 Fw = F1 * cos (30Âş) = 80*cos(30Âş)  
 And for the second cable the component to east is:
 Fe = F2 * cos (20Âş) = F2*cos(20Âş) 
 Then to get a movement northward we need than Fw = Fe, so:
 80*cos(30Âş) = F2*cos(20Âş)
 F2 = 80*cos(30Âş)/cos(20Âş) 
 F2 = 73.73 N  So the magnitude of the east cable should be 73.73 Newtons