At what angle relative to the horizontal should a 52 m curve be banked if the road is covered with ice (no friction) to prevent the car from slipping when traveling at 12 m/s? (use g = 9.8 m/s2)

If there is no friction and the curve is banked, then the only forces present are the normal force and the weight. The normal force is perpendicular to the surface, so it does not contribute to the centripetal force parallel to the surface. Then the centripetal force equals the component of weight that is parallel to the banked surface. If the surface is at an angle θ above the horizontal, the component of weight parallel to the surface is

mg sinθ = mv

^{2}/rg sinθ = v

^{2}/rθ = sin

^{-1}[v^{2}/(rg)] = sin^{-2}[12^{2 }/ 9.8(52)] ≅ 16.4°