A sphere and cone have equal radii and equal volume. Find the relation between the radius and the height of the cone
To find the relation between the radius and the height of the cone with equal volume to a sphere, we can start by considering the formulas for the volume of a sphere and a cone.
The volume of a sphere is given by the formula:
V_sphere = (4/3) * π * r^3,
Where V_sphere represents the volume of the sphere and r is the radius.
The volume of a cone is given by the formula:
V_cone = (1/3) * π * r^2 * h,
Where V_cone represents the volume of the cone, r is the radius, and h is the height.
Since we are comparing the volumes of the sphere and cone, we can set both formulas equal to each other:
(4/3) * π * r^3 = (1/3) * π * r^2 * h.
We can simplify this equation by canceling out π and dividing both sides by (1/3):
4 * r^3 = r^2 * h.
Next, we can divide both sides by r^2 to isolate h:
4 * r^3 / r^2 = h.
Simplifying further, we have:
4r = h.
Therefore, the relation between the radius (r) and the height (h) of the cone with equal volume to a sphere is:
h = 4r.