What is the smallest force F exerted eastward and upward at an angle of 350 with the horizontal that can start the box in motion?

Hey Sierra -- we've got this nearly 3-4-5 /37 right triangle again ...

the eastward/upward force will distribute ~60% up and ~80% east ...

the up part will "lighten" the normal force on the floor: Fn = 150N less 0.6 up ...

F east must overcome 40% of Fn or F east = 60N minus 0.24 up ...

from approx. 3-4-5 tri -> F east is 145% of up => 1.45+0.25 up = 60N ...

up is 60/1.70 or 360/10 or 36N (3x12), making F east (4x12) =>

**F min (5x12)****or F min ~60N ...**Best wishes, ma'am :)