A spherical scoop of vanilla icecream with radius of 2 inches is dropped onto the surface of a dish of hot chocolate sauce. As it melts, the icecream spreads out uniformly forming a cylindrical region 8 inches in radius. Assuming the density of the icecream remains constant, how many inches deep is the melted icecream? Express answer as a common fraction.

volume of icecream = (4/3)π(2^3) = 32π/3 cubic inches

It forms into a cylinder whose radius is 8
let its height be h
π(8^2)h = 32π/3
h = 1/6 inches

To find the depth of the melted ice cream, we can consider the volume of the original spherical scoop and the volume of the cylindrical region it spreads out to.

The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.

Given that the radius of the spherical scoop is 2 inches, the volume of the scoop is V_scoop = (4/3)π(2)^3 = (4/3)π(8) = (32/3)π cubic inches.

The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.

Given that the radius of the cylindrical region is 8 inches, we need to find the height (depth) of the melted ice cream, h_cylinder.

Since the ice cream melts and spreads out uniformly, the volume of the cylindrical region is equal to the volume of the original spherical scoop. Therefore, (32/3)π = π(8)^2h_cylinder.

Canceling out π on both sides gives us (32/3) = 64h_cylinder.

Dividing both sides by 64 gives h_cylinder = (32/3) / 64 = 1/6.

Therefore, the depth of the melted ice cream is 1/6 inches.

To determine the depth of the melted ice cream, we need to find the height of the cylindrical region formed.

The volume of the original spherical scoop can be calculated using the formula for the volume of a sphere:

V_sphere = (4/3)πr^3

where r is the radius of the sphere (2 inches).

V_sphere = (4/3)π(2)^3
= (4/3)π(8)
= (32/3)π

Since the melted ice cream has spread out uniformly and formed a cylindrical region, its volume can be calculated using the formula for the volume of a cylinder:

V_cylinder = πr^2h

where r is the radius of the cylinder (8 inches) and h is its height (the depth of the melted ice cream).

Since the density of the ice cream remains constant, the volume of the melted ice cream is equal to the volume of the original sphere.

V_sphere = V_cylinder

(32/3)π = π(8)^2h

Canceling out π on both sides, we have:

32/3 = 64h

Dividing both sides by 64:

32/3 ÷ 64 = h

h = 32/192

Simplifying the fraction:

h = 1/6

Therefore, the melted ice cream is 1/6 inches deep.