Let's see if I can guess what it is you are doing.
You have a Right triangle.
The legs are known to be 3 cm and 4 cm, making the hypotenuse 5 cm.
Spin this triangle using the hypotenuse as the axis of rotation.
Find the volume of the object so defined.
First: "c^2=a^2+b^2=5" What does that mean? I think it's you finding the length of the third side of the right triangle, but one of those "="s is not telling the truth.
Second: Your 3D object is not a cone. Maybe that is what you meant by "2 Cone". It is 2 Right Circular cones, connected at the base, pointing opposite directions.
Third: The volume of a right circular cone is (1/3)*pi*r<sup>2</sup>*h. I think you have that, but you introduced 'x' and 'y' without telling us what they are. Write definitions clearly. It will help organize your thoughts.
x = the height of one cone.
y = the height of the other cone = 5-x
It looks like you are splitting up the hypotenuse to find the heights of the cones. That is good. Unfortunately, I don't see where you explicitly solved for 'x' or 'y'. On the other hand, maybe you didn't need to do that, since the JOINT volume formula can be factored, giving
(1/3)*pi*r<sup>2</sup>*x + (1/3)*pi*r<sup>2</sup>*y =
(1/3)*pi*r<sup>2</sup>*(x+y) =
(1/3)*pi*r<sup>2</sup>*5 =
(5/3)*pi*r<sup>2</sup>.
Fourth: What you don't yet know is the RADIUS of the cones. Your method to find the height/radius by equating areas is ingenious. I did it a harder way. Draw an altitude to the hypotenuse on your original right triangle. Label the length of the altitude 'h'. If x > y, we have
4<sup>2</sup> = x<sup>2</sup> + h<sup>2</sup>
3<sup>2</sup> = y<sup>2</sup> + h<sup>2</sup>
or
4<sup>2</sup> = x<sup>2</sup> + h<sup>2</sup>
3<sup>2</sup> = (5-x)<sup>2</sup> + h<sup>2</sup>
One can solve for h = 12/5, which is what you have.
This gives the exact answer:
(5/3)*pi*(12/5)<sup>2</sup> =
(12<sup>2</sup>/(3*5))*pi =
((12*4)/5)*pi =
(48/5)*pi =
9.6*pi = <== Your answer and ½ the book's answer
30.159 <== Soroban's answer
It seems like you managed the correct answer, but I found your work exceptionally difficult to follow. Three times, I found myself thinking that you were wondering off, but on later examination, you did what I expected you to do, I just couldn't tell at first.
Write more clearly and if you are missing something, I missed it, too. Don't be afraid to argue with the book, if need be, but it will take a clear demonstration to provide a different answer with confidence.
Edit: Hey, Soroban, you beat me to it, this time.