Final answer:
The radius of the base of the cone is approximately 3 cm. The vertical angle of the cone is the same as the angle of the sector, which is 216°.
Step-by-step explanation:
When a 216° sector of a circle with a 5 cm radius is folded into a cone, the length of the arc of the sector becomes the circumference of the base of the cone. The formula for the circumference of a circle is C = 2πr, where r is the radius. Since the sector is 216° out of the total 360°, we have a proportion (216/360) * (2π * 5) = 2π * r_base, where r_base is the radius of the base of the cone. Solving for r_base, we find that the radius of the base of the cone is approximately 3 cm.
Regarding part (b), the vertical angle of the cone can be found considering the geometry of the situation. The vertical angle is the angle between the slant heights of the cone, originating from the cone's apex. Since the cone is made by bringing the two straight edges of the sector together, the original 216° angle of the sector becomes the vertical angle at the apex of the cone when the sector is folded.