A spherical shell is formed by taking a solid sphere of radius 20.0 cm and hollowing out a spherical section from the shell's interior. Assume the hollow section and the sphere itself have the same center location.

a) If the hollow section takes up 90% of the total volume, what is its radius?

b) What is the ratio of the outer area to the inner area of the shell?

a) To find the radius of the hollow section, we need to find the remaining volume after removing the hollow section.

The total volume of the shell is given by the formula for the volume of a sphere: V_total = (4/3) * π * r_total^3, where r_total is the radius of the complete sphere.

Since the hollow section takes up 90% of the total volume, the remaining volume is 10% of the total volume. So, V_remaining = 0.1 * V_total.

The volume of a sphere is directly proportional to the cube of its radius. So, V_total = V_remaining + V_hollow, where V_hollow is the volume of the hollow section.

Now, substituting the formulas for V_total and V_remaining, we get:
(4/3) * π * r_total^3 = 0.1 * (4/3) * π * r_remaining^3 + (4/3) * π * r_hollow^3.

Since the hollow section and the sphere have the same center location, the radii can be related as follows:
r_total = r_remaining + r_hollow.

Substituting this relation into the equation above, we get:
(4/3) * π * (r_remaining + r_hollow)^3 = 0.1 * (4/3) * π * r_remaining^3 + (4/3) * π * r_hollow^3.

Simplifying this equation, we can find the value of r_hollow, which is the radius of the hollow section.

b) The ratio of the outer area to the inner area of the shell can be found by dividing the surface area of the outer shell by the surface area of the inner shell.

The surface area of a sphere is given by the formula: A = 4 * π * r^2, where r is the radius.

So, the outer area (A_outer) of the shell can be calculated using the radius of the complete sphere (r_total), and the inner area (A_inner) of the shell can be calculated using the radius of the hollow section (r_hollow).

The ratio of the outer area to the inner area is given by: A_outer / A_inner = (4 * π * r_total^2) / (4 * π * r_hollow^2).

Simplifying this equation will give us the answer.