# Water flow graphs

One type of problem you will probably come across is to
sketch a graph showing how water flowing at a *constant rate* fills up
containers of various shapes. For instance, if I had a rectangular prism type
shape, the graph of water height versus time would look like this:

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Because this prism has the same *cross section* all
the way from the bottom to the top, the rate at which the height of the water
would rise would be constant. The *rate* at which the water level rises
is represented by the *slope* of the graph. If the rate is constant, the
slope stays the same – the line is a *straight* one. If the rate
increases (the water level starts going up more quickly), the slope becomes
steeper. If the rate decreases (the water level starts going up less quickly),
the slope becomes less steep. For this case, a constant rate means a *straight
line* on a graph, like the one here.

However, there are more complicated container shapes for which the graph isn’t so simple to plot. For instance, take the example of a cone:

When the water first starts flowing into the tiny hole at the top, it will fill up the bottom of the cone. The bottom of the cone has a large cross sectional area, so the height will go up slowly at first.

When the water reaches the halfway mark, it will be filling
at a much quicker rate. Halfway up the cone, the diameter of the cross
sectional area has *halved*. Or we could just as easily say that the *radius*
of the cross sectional area has halved. Now, because the area of a circle
relies on the *square* of the radius, halfway up the cone the *area*
of the cross section will be a *quarter* the area of the base. So when
the water level is at the halfway mark, the water height will be rising *four
times as fast* as at the start.

So instead of being a straight line graph, the line will start sloping upwards slightly (to represent the water height increasing gradually). Then, as time passes and the water height gets higher, the line will slope more and more steeply upwards, to represent the fact that the water height is increasing more rapidly:

If the cone was upside down, we’d have the opposite of this graph – at first the water height would rise very quickly and then it would slow down:

You can also have combination shapes. It’s best to analyse what the graph would look like for each bit of the shape, and then join the lines together. For instance, this shape:

The bottom of this shape has the largest cross section, so the water level would rise slowly at the start. Then, as the water level rose in the bottom shape, the cross sectional area would reduce, and the rising speed would increase. At the top of the pyramid part, the rising speed would be fast.

The next question is – what happens to the rate when the
water’s rising up in the rectangular prism section? Well, the cross sectional
area of this section is constant all the way to the top. It’s also the *same
size* as the very top of the pyramid part. This means the rising rate will
be the same throughout this section as it was at the *end* of the pyramid
section:

One thing to be careful with is to make sure that you get
the *transition* point at the correct location. The transition point is
the spot on the graph representing when the water level leaves one part of the
shape and starts filling up the next one. Looking at the shape diagram, it
looks like just over half of the total shape’s height is in the pyramid. Just
under half of the pyramid’s height is from the rectangular prism on top of the
pyramid. This means the transition point should be just over halfway up the
vertical axis of the graph: