Could someone help me out?

Question

A civil engineer has been asked to design a drainage system to reduce flood damage during the rainy reason. She produced a fancy design with a triangular cross section as shown in Figure 2. Two months after the structure was completed, there was a heavy rainfall and water gushed into the drain at the rate of 5m3/s. If the drain is 10m deep, 50m long and 8m wide across the top. At what rate is the depth of the water increasing at the instant that the water depth is 6m?

Answer 1

I am not that good at diagrams but if you draw an upside down triangle on the xy axis with the tip at (0,0), you can get the relationship of the volume filling in the triangle. I am assuming that 5m³/s is flowing into the drain but not coming out the other end, so the change in volume is dV = 5 m³/s.

If you do similar triangles, you find that x/y = (4/10) or x = (4/10)y,

Therefore, the cross sectional area in half of the big triangle is (1/2)(.4y)(y) = .2y² and the volume in this half section is 50(.2)y² = 10y²

Now let half of the big triangle fill up with 2.5 m³/sec. then Volume (V) = 10y² and dV = 20y dy

You want to know the change in y when y = 6.

So, dy = dV/20y = 2.5/(20*6) = 1/48 m/sec