Filling a Pond with Water

Hello all you math lovers out there! So I am no longer in gr. 10 and sadly am not doing really super, hard math anymore but I thought I would go through the calculations for the math project I just handed in today :D There are a few things that you need to know but I'm too tired to fully explain, so it's going to be as simple as I can make it. I'll begin with diagrams, as I know everyone loves diagrams :)

Figure 1 - a top view of the pond area that you are trying to fill, the depths of each of the dots are given in the question.

Figure 2 - profile of the river that will be used to fill the pond, each depth is given in the question.








.
Figure 3 - table of the depths, etc of the pond in Figure 1

The project was to find out how long it took for a pond to fill with only rainwater, how long it would take to fill the pond if a nearby creek was diverted to the pond and the total volume of the pond. Figure 3 shows the first table I set up in excel to determine the volume of the pond. 

I was giving the depths of the pond in different points across the pond and with that you use either the Trapezoid rule or the Simpson's rule . 

Simpson's rule: h/3(y0+4*y1+2*y2+4*y3+2*y4+ . . . +2*y(n-2)+4*y(n-1)+y(n)
Trapezoid rule: h/2(y0+2*y1+2*y2+2*y3+2*y4+ . . . +2*y(n-2)+2*y(n-1)+y(n)

Using either the Simpson's or the Trapezoid rule you can determine the surface area of each of the cross sections of the pond. Then with all those areas you can use the Simpson's rule again to determine the volume of the pond! I set my excel spreadsheet up like this.

Figure 4
So the total length of the pond <1055m> divided by the number of sections <6> gives you height <~176m> and then just plug that number into the Simpson's rule along with all the previous numbers you calculated on the far right side of Figure 3. This means that the total volume of the pond is ~869633m^3!

Next you determine the "flow rate" of the rain that will fall on your area that will collect in the pond so that you can calculate the rough time it will take to fill the pond with only rain as the input.

Figure 5

The average rainfall on the area was 260mm/year. Once you go through the conversion factors to get into m/year you can use that number in your flow rate calculation as your speed. Given a surface area of 208hectares, you go through more conversion factors to get to m^2 and use this in your flow rate equation as surface area.

Symbols:

Q=Flow Rate
A=Surface area
s=Speed
V=Volume
t=Time

Flow Rate Formulas:
Q=As
Q=V/t

You must rearrange the second flow rate equation so you end up with time as the desired variable. Plugging in the correct volume and flow rate numbers gives you a time of ~1.608 years but since the questions asks for days you simply multiply that number by 365 and you get ~587 days. Yay, we know have the answer to our first question.

Next we have to calculate the surface area of the river that could be diverted. You use the Trapezoid rule in this case as there are 5 sections.

Figure 6 - table showing the depths, etc of the river in Figure 2

This shows that the surface area of the river is ~1.73m^2.

The average speed of the creek is given as 5.53cm/s so we must convert that number into m/year either now or later, but I did it here just to make things easier.

Figure 7

Flow rate of the river is just surface area times speed. Then you use the volume of the lake as V and you must add the flow rate of the river and the flow rate of the rainfall to get total Q. In conclusion, Figure 7 shows that, when the equation of Q=V/t is rearranged to t=V/Q you find out that t is equals to ~0.244years. Since the question asks for days, you once again multiply the number by 365 and we get ~89days.

Therefore: the lake, being 869633m^3, would take 89 days to fill if the creek was diverted and 587 days to fill if the creek was not diverted.

If you find any errors in my calculations, good for you, your smart, please don't tell me as I never want to look at this question again :) Have fun interpreting my mess of numbers and bad explanations.

-Eleanor