Question

A spherical shell whose inner radius is 2 cm and whose outer radius is 8 cm is made of material of varying density. At a point {x, y, z}, the density of the material is 0.39 times the square of the distance to the center of the sphere. Assume all the air has been pumped out of the hollow interior. Does the shell sink or float in water?

Answer 1

Density is 16.11 units, If it is less than the density of water in the same units, the sphere will float.

Explanation:

We have to:

Mass of shell at distance r from:

Density = mass / volume, if we solve for mass we have:

Mass = volume * density

replacing:

Mass = (4 * pi * (r ^ 2) * dr) * (0.39r ^ 2)

Mass = 1.68 * pi * r ^ 4 * dr

Mass = 1.68 * pi integral from 2 to 8 of {r ^ 4 * dr}

Mass = 1.68 * 3.14 * (r ^ 5) / 5 when r equals 2 to 8, replacing:

Mass = 1.68 * 3.14 * [(8 ^ 5/5 - (2) ^ 5/5]

Mass = 34537.79

Now the volume would be: (4/3) * pi * r ^ 3, we replace:

(4/3) * 3.14 * 8 ^ 3 = 2143.57

Density = 34537.79 / 2143.57 = 16.11 units

If it is less than the density of water in the same units, the sphere will float.