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.








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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

Problem Solving Set #3

7. In this diagram, square ABCD has side length 2, with M the midpoint of BC and N the midpoint of CD. The area of the yellow region BMND is. . .

It took me a few tries before I got the answer to this question, but once I realized what to do, it was pretty easy. First you have to find out the area of the triangle ABD. The formula for triangles is side time side divided by 2. So, just plug it in and you get 2 times 2 divided by 2. That gives you... 2! Yeah! So the area of triangle ABD is 2.


Now you have to find the area of angle NMC! Before you do this you have to figure out what sides NC and MC are. By reading the question again you see that N and M are midpoints; meaning that on each side of these points is 1... because 1 plus 1 equals 2! So now you know that MC and NC both equal 1. By knowing this we can plug it in and we get 1 times 1 divided by 2. This equals... 0.5! YAY!


Sadly we have not quite finished the question so we have to rap it all up by actually finding the shaded area. We take triangle ABD and triangle MNC and minus them from the entire square which is 4. This will look like 4 minus (2 + .5) = 1.5. There now you`re done. 1.5. That's all there is to it.


The area of the shaded region of BMND is (D) 1.5


Well actually the answer puts in in 3 over 2 form, but it's the same thing. 3 over 2 is the same as 1.5

Problem Solving Set #2

For this set of questions I will pick number 16 to show. I like this question because, you can use math or just common sense ! Here's the question.

16. On a rectangular table 5 units long and 2 units wide, a ball is rolled from point P at an angle of 45 degrees to PQ and bounces of SR. The ball continues to bounce of the sides at 45 degree angles until it reaches S. How many bounces of the ball are required?

The first thing I did was read the question a few times to make sure I knew what I was supposed to be looking for. Then I looked at the rectangle given and tried to figure out where to start. Then when you realized that each bounce is 45 degrees and it's starts at P I started drawing the path of which the ball would've taken. This is one answer.

Or you can mathematically deduct that each roll between the bounce off another wall will be 2 units, except for the end where there is not enough space. There is only enough room for 5 bounces, no more, no less.

So the answer = (D) 5

P.S. Everything in the picture that is black was given with the question. Everything in Red (or whatever that colour is) is what I drew to help figure it out.

Monster Equations!

When I first look at these questions, I think, yeah right, I can't do either one of these... but as soon as you start breaking them up into parts and looking at it separately, it becomes easier.

1a) First you put all of the negative exponents on the reciprocal top or bottom so that they are positive.
b) Spread out the 5 over 2 to the entire equation.
c) Solve each root separately.
d) Make sure they are in their simplest form!
Answer) Write down your answer and put it in a box so it is noticeable!

2a) Change the roots to x over y as exponents of the number.
b)Solve each section of the equation separately.
c) Take out any common numbers to put in simplest form
Answer) WRITE OUT YOUR ANSWER!

I don't know how to use the Word Equation something-something, so, I hope this helps anyways!

What does it mean to be good in math?

Today in Math class we were trying to find qualities that would make a person good at math!
We came up with a long list and now I have to pick three of the ones I think are the most important. These are the three - hard working , willing to learn and you must have perseverance .

You must be hard working because in order to solve a problem, sometimes you have to keep working at it - sometimes longer than you want to. When you don't understand something, you can't give up and forget about it, you have to keep trying to find the answer. Whether it's looking at the question in a different way, or going back to the basics of the question - you have to find it! Else it will bother you for the rest of your life!

Being willing to learn is a very important characteristic in math because there is ALWAYS more to learn. Math keeps going and you can't just want to stop after learning a certain amount... you have to want to learn more. You also have to be willing to learn from your mistakes. It's not a bad thing when you make mistakes because the next time you come across that question, you'll remember the mistake you made and you'll remember how to do it properly.

You also must have perseverance. This is obvious, because you can't just stop in the middle of a question because you don't know what to do - look back over your work, see if you did anything wrong or can do something differently

Problem Solving Set #1

On February the 9th my class recieved a worksheet to finish for homework. Sadly I did not get 100%, but they say that you learn from your mistakes, so if you never make a mistake, you won't learn.

So, the question I will go over is #11. Twelve balloons are arranged in a circle as shown. Counting clockwise, every third balloon is popped. C is the first one popped. This continues around until two unpopped balloons remain. The last two remaining balloons are: (A) B, H (B) B, G (C) A, E (D) E, J (E) F, K

Starting at C, you cross out every third balloon clockwise. It should be in a pattern like this: C, F, I, L, D, H, A, G, B, K. Once you get to K, you should only have 2 balloons left: E and J!

This is the only way I know to solve this problem and it's really easy too! When I was solving it I wanted to play MASH!

So, I hope now you understand this question and it all makes sense to you !

Tower of Hanoi

Describe the strategy and the formula for the puzzle: Tower of Hanoi

When I first played this game, I used the method of guess and check ... but after I got to 5 disks and couldn't find an answer, I got incredibly bored and frustrated .

Seeing as I couldn't find a mathematical solution , I'm still currently using the lovely method of guess and check!:)

My picture above proves to you that I have beaten 5 disks and if you can't do this then maybe my next piece of information will be helpful! All I've found out so far is, if you have an odd number of rings , you have to start by putting the smallest ring on the furthest pole and the second smallest ring on the middle pole. For an even number you would need to reverse this - putting the smallest ring on the middle pole and the second smallest ring on the furthest pole.

I don't know how else I can help except, don't give up, because it's problems like this that we're happy to solve! I can tell you I was... after looking at a few hints... But anyways, DON'T GIVE UP! One of the first rules in math, you are not allowed to say "I can't," because else we'll have to kick you out!