The Water Container 22April 2021
Questions Generated
Taken from the book Mathematical Modelling; Applications with Geogebra; Pg 247
Please follow following steps in your own geogebra file to start understanding the problem (the steps shown below are carried out above also, for ease).
We wish to observe how change in height of water within a dispenser bottle affects the flowrate of water coming out of it. We are assuming here that the water level in the dispenser labelled y can be modelled by the differential equation y' = -k sqrt(y) for some constant k. Try finding out why?
What is your observation or hunch? Try pouring out water from your dispenser when it is full; try the same when it is say, a quarter full; is there any difference? Can we try and predict what change occurs in its flowrate once it is quarter full, compared to when it is full? Lets give it a try!
1) Create a point (0,h) where h will become a slider with height h; it will be the initial height of the water;
2) Create a slope field by typing in the command: slopefield[-ksqrt(y)]
3) Create a solution through the point A with: solveODE[-ksqrt(y),A]
4) Create a horizontal line y= 0.25h
5) Find a new intersection point between this line (y=0.25h and the graph obtained by step 3 above.
6) Put a random point named D on the graph;
7) Use this command: fitpoly[A,B,D,2]; and you will get almost the same graph as obtained in step 3.
8) Create the derivative v(x) = f'(x)
9) Input v0 = v(0) in algebra input bar followed by v25=v(25), to create the speeds for the full and the quarter dispenser.
10) Find out what you get by inputting v25/v0.
After making curve,
Put k=0.9 and choose h=10,
1) How much time does it take to fully empty the dispenser, considering it is full (h=10)
2) How much time does it take to empty the dispenser when it is 25% filled only?
3) Change slider h; what do you observe?
4) What is v=f'(x) giving us? Compare v(0) and v(25) and their significance.
5) Change values of k and h and find out what is changing. Can you describe the changes? What is k and what is h?
6) What does the slopefield give us?
NEW QUESTIONS TO WORK ON:
1) You observe in this question that we are given a differential equation already. Can we come up with the same or similar differential equation ourselves; by say carrying out an experiment at home ourselves? So this will be a scenario where we may not have this differential equation to start with?
2) Plot y=x^2; compare with your original problem equation - They are very similar. What are the differences?
3) Take solve ODE of both curves. What are their solutions and compare them.
4) What can i do to hide the right half of this graph as container empties only in our case we need to omit the right half of the graph; can you help me?
5) fitpoly command (we used three points - what is result if we take only two points and use this command?6) Is there a difference between the solveODE and fit poly results? solveODE gives us the conic section?
A big statement: Why don't you go ahead and find out by yourself whether this is true or not that:
Regardless of values of k and h, you will observe that a glass will always take twice as long time to fill when the dispenser is quarter full compared to when it is full! Why not check this out by yourself!