A large waffle cone has a height of 7.25 inches and a radius of 1.5 inches. Two scoops of sorbet are placed on the wide end of the cone. The scoops are spheres with a radius of 1.5 inches. If all the sorbet melts into the cone, will the cone overflow? Show all of your work.
To determine if the cone will overflow, we need to find the volume of the cone and the volume of the sorbet.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Substituting the given values, we get:
V_cone = (1/3)π(1.5^2)(7.25)
V_cone ≈ 15.31 cubic inches
The volume of a sphere is given by the formula V = (4/3)πr^3.
Since we have two scoops of sorbet, we need to find the total volume by doubling the volume of a single scoop:
V_sorbet = 2[(4/3)π(1.5^3)]
V_sorbet ≈ 14.14 cubic inches
So the total volume of the cone and the sorbet is:
V_total = V_cone + V_sorbet
V_total ≈ 29.45 cubic inches
Since the total volume is less than the volume of the cone alone, the sorbet will not cause the cone to overflow.