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.

Volume = pi*(radius^2)*height/3
Surface = pi*radius*(sqrt[radius^2 + height^2])
       
Let X = radius = height.

Volume = Surface
pi*(X^3)/3 = pi*X*(sqrt[2X^2])
x^3/3 = X*[sqrt(2X^2)]
(x^6)/9 = (X^2)*(2X^2)
(X^6)/9 = 2X^4
X^6 = 18X^4
X^2 = 18
X=sqrt(18)

radius = height = sqrt(18)
Volume = Surface = given by either formula, above

The other possibility is the "degenerate cone": radius=height=volume=surface area = 0. When I was a kid, my parents were so mean, they only gave me "degenerate ice cream cones" during the summer.   

 

 

 

 

 

Edited on October 11, 2004, 10:29 am

Posted by Penny on 2004-10-11 10:11:10