A solid cone of height 12cm and radius 9 cm is recast into a solid sphere. Calculate the surface area of the sphere.
cone V= 1/3 pi r^2 h
sphere V= 4/3 pi r^3
area of sphere = 4r^2
how to find sphere of radius?
To find the radius of the sphere, we need to determine the volume of the cone and equate it to the volume of the sphere.
The volume of the cone is given by V_cone = (1/3) * π * r^2 * h, where r represents the radius of the cone and h represents the height of the cone.
Substituting the given values, we have V_cone = (1/3) * π * (9 cm)^2 * 12 cm.
Simplifying this equation, V_cone = 108π cm^3.
Now, equate the volume of the cone to the volume of the sphere. The volume of the sphere is given by V_sphere = (4/3) * π * r^3, where r represents the radius of the sphere.
Hence, we have (4/3) * π * r^3 = 108π cm^3.
To solve for r, divide both sides of the equation by (4/3) * π.
This results in r^3 = (108π cm^3) / ((4/3) * π).
Simplifying further, r^3 = 81 cm^3.
To find the cube root of 81 cm^3, take the cube root of both sides of the equation.
Thus, r = ∛(81 cm^3).
Evaluating this expression, r = 4.3267 cm (rounded to four decimal places).
Therefore, the radius of the sphere is approximately 4.3267 cm.