leakingbottle

=The Leaking Bottle := =Cheryl Chau, Shirley Hsieh, Chloe Lin, Kelly Robinson=

The goal of this investigation is to model the rate of leaking water from a hole in a container. This was done with a bottle with a hole, water, team members to do different tasks, and timer. By calling out each 10 second intervals, we recorded the height of the remaining water in the bottle. Initially, the water started out at 15cm, over time, it eventually got down to 0.5.

Trial 1 Trial 2
 * Time (secs) || Water left (cm) || Time (secs) || Water left (cm) || Average (cm) ||
 * 0 || 15 || 0 || 12 || 150 ||
 * 10 || 12.2 || 10 || 12.1 || 121.5 ||
 * 20 || 9.3 || 20 || 9.5 || 94 ||
 * 30 || 7.2 || 30 || 7.2 || 72 ||
 * 40 || 5.4 || 40 || 5.2 || 53 ||
 * 50 || 3.8 || 50 || 3.7 || 37.5 ||
 * 60 || 2.5 || 60 || 2.5 || 25 ||
 * 70 || 1.5 || 70 || 1.5 || 15 ||
 * 80 || 0.6 || 80 || 1.1 || 9 ||
 * 90 || 0.5 || 90 || 0.6 || 5.5 ||

This is what the data looks like. By using finite difference, it's possible we can obtain a function for this data. We used the finite difference in the calculator. When we did the first difference, we found that the data showed a pattern:

Then we did the next difference, but the data did not clutter nicely around the x-axis.

Then we tried the next one, and found that the data plotted nicely around the x-axis. This meant that this polynomial had a degree of 3.

On top of that, we did it by hand :

6a= 0.5 24a+2b= 3.5 37a+7b+c= -19 27a+9b+3c+d = 72

With that, we found the values of those letters: a= 0.0833 b= 0.7504 c= -27.33 d= 150

Which means the equation to model this data is : y= 0.0833x^3 + 0.7504x^2 - 27.33x + 150

We checked this with the regression on the calculator which was : y= -1.67E -5 x^3 + 0.0186x^2 -3.149x + 150



The equation found by hand was definitely way off the data, unlike the equation we found from the calculator.

After consulting with our teacher, we discovered that the reason our equation was way off the data was due to the fact that the equations we used only worked with x increasing consecutively by one. Our data's x points increased by 10's. Therefore, it was suggested that we should use the matrix method instead.

First we had to lay out four equations in order to figure out the variables a,b,c,d: 150= a(0)^3 + b(0)^2 + c(0) + d (so d is obviously equal to 150) 121.5= a(10)^3 + b(10)^2 + c(10) + d 94=a(20) ^3 + b(20)^2 + c(20) + d 72=a(30)^3 + b(30)^2 + c(30) + d

Then we rewrote this into a 4 x 5 matrix into [A]: 0 0 0 1 150 1000 100 10 1 121.5 8000 400 20 1 94 27000 900 30 1 72

Then we inputed rref([A]) into the calculator to determine what a,b,c,d was. The results were as following: 1 0 0 0 7.5 x 10^-4 0 1 0 0 -.0175 0 0 1 0 -2.75 0 0 0 1 150

Therefore: a= 7.5 x 10^4 b= -.0175 c= -2.75 d= 150

The equation would be: y= (7.5x10^-4)x^3 + (-.0175)x^2 + (-2.75)x + 150 Thus, looking like this : Still, the calculator equation is better than our equation found by hand. Thus, we stick with the calculators regression.

Using Torricelli's Law: **Torricelli's Law** states that the speed of a fluid flowing out of an opening under the force of gravity is proportional to the square root of the product of twice the acceleration of the gravity multiplied by the height //h//, the distance between the level of the surface and the center of the opening.

v= sqrt (2//gh// ) //g//= 9.8m/s //h//= .15m Which equals: 1.715 m/s This speed coincides with the speed the fluid would have in a freefall from the height //h//. Thus, if the hole was at the bottom, it would be in that speed. However, the result is logically incorrect as the water in the bottle doesn't flow out at 1.715 meters per second. Therefor, we hypothesize that the diameter and size of the hole in which the water flows out must be taken account for. In conclusion, the answer derived from Torricelli's law is incorrect.

In conclusion, the best polynomial that would model the rate of the water coming out of an opening near the bottom of the bottle is a 3 degree polynomial.