An ice cream cone has a height of 6 inches and a radius of 2 inches. A scoop of ice cream sits on the top of the cone. The scoop is a sphere with a diameter of 4 inches. If the entire scoop of frozen yogurt melts into the cone, will the cone overflow? Show all your work and explain your reasoning.

Yes, it will overflow, because the volume of the scoop of ice cream is higher than the volume of the cone.

Step-by-step explanation:

To know if the cone will overflow when the entire scoop of frozen yogurt melts, we need to compare the volume of the scoop and the volume of the cone: if the volume of the scoop is higher, it will overflow.

The volume of the cone is:

V_cone = (1/3) * pi*r^2*h

Where V_cone is the volume, r is the radius and h is the height. So:

V_cone = (1/3) * pi*2^2*6 = 25.1327 in3

The volume of a sphere is:

V_sphere = (4/3) * pi*r^3

Where V_sphere is the volume and r is the radius. If the diameter is 4, the radius is 4/2 = 2. So:

V_sphere = (4/3) * pi*2^3 = 33.5103 in3

The volume of the sphere is higher, so the cone will overflow.