maths problem: cyclinder

The total surface area of the cyclinder = 2πrh + 2πr^2 = 2πr(r + h)
The volume of the cyclinder = πhr^2
If you know the total surface area and the volume of it ,
you can set up two equations :
A = 2πr(r + h)......(1)
V = πhr^2......(2)
to find r and h .
By (2) :
h = V / (πr^2) , let it be k / r^2 (where k = V / π) , sub to (1) :
A = 2πr (r + k/r^2)
A = 2πr^2 + 2kπ / r
A/2π = r^2 + k/r , Let p be A/2π :
p = r^2 + k/r
pr = r^3 + k
r^3 - pr + k = 0
You need to solve the above cubic formula,
It may not have a positive real root.

2010-04-30 21:59:32 補充：
冰璃 :

Your example is not about total surface area and the volume。

2010-04-30 23:10:46 補充：
For example :

The total surface area of the cyclinder = 12π
The volume of the cyclinder = 5π

then

12π = 2πr(r + h)......(1)
5π = πhr^2......(2)

By (2) :

h = 5 / r^2 , sub to (1) :

12π = 2πr(r + 5/r^2)

6 = r(r + 5/r^2)

6 = r^2 + 5/r

2010-04-30 23:11:03 補充：
r^3 - 6r + 5 = 0

(r - 1)(r^2 + r - 5) = 0

r = 1 or r = 1.79129 or r = - 2.79129(rejected)

When r = 1 , h = 5
When r = 1.79129 , h = 1.55825

More than 1 solution , so the shape of the cyclinder is uncertain.