I'm having some trouble with the following questions:

The vertical cross-section of a bucket is shown in this diagram. The sides are arcs of a parabola with the y axis as the central axis and the horizontal cross-sections are circular. The depth is 36 cm, the base radius length is 10 cm and the radius length of the top is 20 cm.

a)From this part, we know that the parabolic sides are arcs of the parabola $\displaystyle y = 0.12x^2 - 12$.

b) Water starts leaking from the bucket, initially full, at the rate given by $\displaystyle \frac{dv}{dt} = \frac{-{\sqrt{h}}}{A}}$ where, at timetseconds, the depth ishcm, the surface area isAcm^{2}and the volume isvcm^{3}. Prove that $\displaystyle \frac{dv}{dt} = \frac{-3{\sqrt{h}}}{25\pi(h+12)}}$.

I can recognise that $\displaystyle \frac{3}{25}$ represents 0.12, so I know that part (a) ties in somehow, but I don't know where to start.

c) Show that $\displaystyle v = \pi{\int_0^h}({\frac{25y}{3}} + 100)dy$. I don't know how to approach this either, but it will probably become clear once I understand part (b).

This container has an open rectangular horizontal top,PQSR, and parallel vertical ends,PQOandRST. The ends are parabolic in shape. Thexaxis andyaxis intersect at O, with thexaxis horizontal and theyaxis the line of symmetry of the endPQO. The dimensions are shown on the diagram.

a)The equation for the parabolic arc QOP was found to be $\displaystyle \frac{2}{5}x^2$.b) If water is poured into the container to a depth of

ycm, with a volume ofVcm^{3}, find the relationship betweenVandy.

I initially tried to grapple with the idea that the volume = the area multiplied by 60, but to no avail.

Thanks in advance, I know these can be long winded questions.