There is a cone whose radius is equal to its height. The numerical surface area is equal to the numerical volume. Find the exact radius and volume of the cone.

Since the problem stated that the cone had a radius, I assumed this meant it was a circular cone. Technically, this could still be either a right circular cone, or an oblique circular cone. The volumes would be the same either way. I don’t know about the surface areas. However, since the surface area of an oblique circular cone would be harder to calculate, I took the lazy route and chose to find a solution with a right circular cone.

Calling the radius, r, and the height, h, then the formulas for the surface area, S, and the volume, V, are:
S = pi*r[r + sqrt(r^2 + h^2)]
V = pi*r^2*h/3

Since r = h, we get:

S = pi*r[r + sqrt(r^2 + h^2)]
S = pi*r[r + sqrt(r^2 + r^2)]
S = pi*r[r + sqrt(2*r^2)]
S = pi*r[r + sqrt(2*r^2)]
S = pi*r[r + r*sqrt(2)]
S = pi*r^2[1 + sqrt(2)]

V = pi*r^2*h/3
V = pi*r^2*r/3
V = pi*r^3/3
V = pi*r^3/3

V = pi*r^3/3 = S = pi*r^2[1 + sqrt(2)]
r/3 = [1 + sqrt(2)]
r = 3*[1 + sqrt(2)]

V = pi*r^3/3
V = pi*[7 + 5*sqrt(2)]/3

Those are the exact values for the radius and volume. "Approximate" values are r = 7.24264068711928 and V = 397.850069393283

I called my subject "a solution" because I'm sure there is an oblique circular cone that meets these requirements.

Posted by nikki on 2004-10-11 13:07:23