Answer:
The volume of the solid cone is given by the formula:
V_cone = 1/3 * π * r_cone^2 * h_cone
where r_cone is the radius of the cone and h_cone is the height of the cone. Substituting the given values:
V_cone = 1/3 * π * (9 cm)^2 * 12 cm = 1080π cm^3
This volume is then recast into a sphere. The volume of a sphere is given by the formula:
V_sphere = 4/3 * π * r_sphere^3
where r_sphere is the radius of the sphere. Since the volume of the cone is equal to the volume of the sphere, we can write:
V_cone = V_sphere
1080π cm^3 = 4/3 * π * r_sphere^3
Solving this equation for r_sphere, we get:
r_sphere = (810/π)^(1/3) cm
The surface area of a sphere is given by the formula:
A_sphere = 4 * π * r_sphere^2
Substituting the value of r_sphere that we just calculated:
A_sphere = 4 * π * ((810/π)^(1/3))^2 = 4 * π * (810/π)^(2/3)
which simplifies to:
A_sphere = 4 * π * ((810)^(2/3)/(π^(2/3)))
You can use a calculator to get a numerical approximation for this, but this is the exact expression for the surface area of the sphere.