This problem is a generalization of Hypotenuse from Cone
An acute triangle is rotated about each of its three sides to produce three double-cones of differing volumes.
The three double-cones have volumes of 55440*pi cm^3, 47520*pi cm^3, and 36960*pi cm^3.
Determine the perimeter of the triangle.
(In reply to Solution
I see one mistake, you wrote "So b = 166320a/142560 = (5/6)a" This should equal (7/6)a rather than (5/6)a. That threw off the rest of your arithmetic. Other than that you used the same method I devised.
For the numbers, I just picked a 6-7-9 triangle and scaled it up a bit, so yeah there are going to be roots without deliberately choosing a Heronian triangle.