Work done by spring vs hookes law?

[kahwawashay1]

what is wrong with this problem?

http://www.cramster.com//answers-mar-10/physics/buoyancy-force-fig-14-36-spring-spring-constant-360x104-mis_811494.aspx

In Fig. 14-36, a spring of spring constant 3.60x104 N/mis between a rigid beam and the output piston of a hydraulic lever.An empty container with negligible mass sits on the input piston.The input piston has area Ai, and the outputpiston has area 15.5Ai. Initially the spring isat its rest length. How many kilograms of sand must be (slowly)poured into the container to compress the spring by 5.80 cm?

See above link if you want to see image.
Basically I solved this same as was solved in the above link, but I do not understand why Hooke's law is used instead of the kx^2/2. What I did was:

work to compress spring=kx^2/2
work applied to piston 1=mgy₁, where m is mass of sand
the two works are equal by Pascal's Law, so kx^2/2=mgy₁

the volume of water displaced same at both pistons:
V=A_iy_i
V=18A_ix
Solving for y_i and x and plugging into my work equations gives:

(k/2)((V/18A_i)^2)=mg(V/A_i)

So the V does not cancel out. But I cannot see how Hooke's Law can be used. Yes, the spring will feel force of kx once it is compressed 5cm, but the work to achieve this compression is a variable force, so we must integrate or deal with the work...?

 

[rock.freak667]

From what I can gather, if you use conservation of energy, then you'd have to take into account the fact that the beam might move up some distance y₂ while the spring will compress a distance x.

Whereas if you were using Hooke's law, the pressure transmitted throughout the fluid is constant. So the force on the beam is the force that the spring is seeing.

 

[kahwawashay1]

From what I can gather, if you use conservation of energy, then you'd have to take into account the fact that the beam might move up some distance y₂ while the spring will compress a distance x.

y₂=x

The distance the spring compresses is obviously equal to the distance the beam moves up...so that doesn't solve it..