2013 AJC P1 Q5

A trough is \(8\) metres long and its cross-sectional ends are in the shape of an isosceles triangle whose width is \(5\) metres and height is \(2\) metres. It is held in the position as shown in the figure by stands of the same height. At time \(t\) seconds, the height of the water in the trough is \(h\) metres and the width of the water surface is \(w\) metres. The trough is initially empty and water is being pumped in at a constant rate of \(5\) m\(^{3}\) /s.

1. Show that \(w=\frac{5}{2}h\). Hence find the volume of the water in the trough in terms of \(h\).[2]
2. Find the rate of change of \(h\) at \(t=2\).[3]

From \(t=2,\) the rate at which water is being pumped into the trough is changed to \(2h\,{{\mathrm{m}}^{3}}/\mathrm{s}\mathrm{.}\)

3. Find the total time taken for the trough to be completely filled.[3]