Name
Tutor
Course
Date
Mathematics project
Introduction.
Purpose statement.
The reason for choosing this particular station is that it is among the stations with a depth to water level graph with an adequate amount of data. This means that the study will yield robust results and it will be easy to plot the lines of best fit. This station is also among the stations identified with a red triangle on it. The red triangle means that the station has a low depth to water level. The most current measurement for this station is 7.63 which makes the station interesting compared to other stations.
Objectives.
To understand the application of lines, slopes and piecewise functions in real life. Lines are very important in studying the relationship between variables. When the relationship between variables can be defined as linear, it is possible to find it in the same way we find the equation of a line from two coordinate points. Slopes are important since they help as to state how rapidly variables are changing. On the other hand, a piece-wise function helps to express variables with different expressions at different parts of the domain.
This study aims to find out the relationship between depth to water level over time using lines and piece-wise functions.
The study also aims to find out how rapid the change of depth to water level over time has been in the particular station.
The study aims to predict the depth to water level over the next twenty years for the station in question and determine whether there are any potential issues as a result of this prediction.
This study will also try to find out the solutions to any potential issues that may arise from the study.
Location of the station.
The station in this study has a site number 404621111422501. It is located along S. Geneva road and is adjacent to Orem Center Business Park, UT. This location was selected because it is near Orem Business Park, Utah Valley University, Lake Side Sports complex and other public facilities that draw water from this particular station. The reasons why this study considers the facilities neighboring this station is because public supply is one of the major ways which ground water is used.
Body.
Description of the graph.
The graph has both the y-axis and x-axis. In the y-axis to the left-hand side, the depth to water level is represented in feet below the land surface. This y-axis is calibrated with both positive value and negative values. On the right-hand side, the graph has another y-axis which represents the elevation above the National Geodetic Vertical Datum 1929 which is measured in feet. The National Geodetic Vertical Datum of 1929 is the datum established for vertical control surveying by the general adjustment of 1929 in the United States of America. On the other hand, the x-axis represents time in years since data was started to be collected in the particular well station.
Continuous Piecewise function inserted within the graph.
At this point, the lines of best fit are drawn within the graph. These lines will help in the generation of the piecewise function by finding their equations and then expressing them in the piecewise form.
In order to find the slope of the steeper line, the points (97,-5) and (86,-10) are used. In order to find the slope of the other line the points (46, -13) and (86,-8) are used. The slope of the steeper line can be obtained dividing the change in the variable y by the change in the variable x using the points indicated above as follows:
(-5+10)” (97-86) = 5/11.
The slope of the other line can be found similarly as follows:
(-8+13) ” (86-46) = 1/8
The equations of the line are the found using the point slope formula which is shown below:
y-y_1=m(x-x_1 )
The equation of the first line can be found as follows considering the point (86,-10):
y + 10 = 5/11(x ‘ 86)
y = 5/11x ‘ 540/11
The equation of the second line can similarly be found as follows considering the point (86,-8):
y + 8 = 1/8 (x – 86)
y = 1/8x ‘ 75/4
Now, in order to obtain the piecewise function the lines of best fit in the graph above are observed and it is noticed how the distribution of data changes at the point x is equal to 86. This implies that this point shall be used as the boundary of the two lines of best fit to form the piecewise function.
Using the above equations the continuous piecewise function can be defined as a single function as follows:
{‘(f(x)=5/11 x-49.1,x<86@f(x)=1/8 x-18.75,x’86)’
Conclusion.
Projecting twenty years into the future, a new y-axis is placed in the year 2000 in order to make work easier. This makes the y-intercept be -4. This can be shown in the equation below:
f(x)=1/8 x+4
Twenty years into the future makes the value of x to be 36. As a result, the prediction will yield the following:
f(x)=1/8 (36)+4= 8.5
The above results mean that in twenty years, which is in the year 2036, the depth to water level will be 8.5. On the other hand, the National Geodetic Vertical Datum can be obtained through extrapolation. The ground water elevation level above the National Geodetic Vertical Datum will be at 4519 feet.
Potential issues.
The potential issues that arise are related to ground water withdrawal. This is because all water that is discharged by a well comes from a loss of water within the system. This means that there may be a decreased normal water flow from the area. What happens in most cases is that when the pump drawing water from the well is not set deep enough to allow a potential decline in water level and the water level has dropped below the pump's intake then the problem of water shortage arises. The water level may drop because of drought conditions or due the fact that the well may be shallow or the fact that the amount of water being drawn from the well has increased.
Another potential issue that may arise is one of increased water levels. The rise in water levels can be explained as a result of curtailed pumping of water in areas where there has been a large ground-water withdrawal. Such areas could be areas of mine dewatering, municipal water supply which kept the water table below the known natural levels. As a result of curtailment of water pumping for a long time, the water table rises to the original natural level. This may then result in the flooding of underground structures that were built when the water table was lowered.
Basement flooding and foundation failures are other potential issues that may arise. This occurs from the fact that in shallow aquifers, water levels are lower artificially over large areas. As a result of this, individuals end up constructing houses in these areas since they are now dry and if the water level resumes its original levels, then basement flooding and foundation failures occur. This occurs especially where the natural water level of the land surface is within ten feet. When this occurs the only solution is resuming pumping in order to restore the groundwater levels to a favorable level. In the same way, some houses were built in periods when there was extended drought, at this time the water tables was low and when the wet period returned the water table rises and floods basements.
Close monitoring of water levels is important. This is because these records may show whether high water levels are only related to climatic events or whether there is a recovery of water levels as a result of ceased pumping in areas that are nearby. Many areas have an increased water withdrawal for mining or industrial uses. This could cause serious issues in the future once ground water withdrawal ceases.
Preventive measures.
In order to avoid ground water depletion, preventive measures can be taken. These may include avoiding continuous running of water while in use. This could be when brushing teeth or shaving, you should turn off the tap to avoid water wastage. Another measure could be to fix any dripping water facilities. You should make sure to check the toilets and taps at home for any leaks and get them fixed. Installing water conserving models in homes and in places that require water usage come a long way to reducing groundwater depletion.
Ground water depletion can be prevented or reduced through watering plants wisely. This is because a great percentage of ground water usage goes to irrigation. Therefore, watering plants should be done when the plants really need it and it should be done during the coolest parts of the day so as to avoid wastage of water through transpiration.
Another way to reduce ground water depletion is through education. Individuals should get involved in water education and learn more about groundwater and the various ways to prevent depletion. Most importantly, they should share this information with others in order to achieve maximum effectiveness in saving water.
When it comes to prevention of rising of water levels, there are no many preventive measures. However, a background check should be done on the water table level in previous years before constructing houses in order to avoid basement flooding and foundation failures.
Lessons learned from this project.
A lot has been learned from this project. The lessons learned include the application of slopes in real life. This can be seen when slopes have been applied in measuring how rapidly the level of ground water has changed in the station studied. Much has also been learned in the application of lines where linear equations are obtained. These equations describe the relationship between variables. The project has also helped learn the application of piecewise equations and how different expressions are used to express a function whose domain is divided into different parts.