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Please show all work details with answers, insert the graph, and provide answers to all of the critical thinking questions on this document.
Mathematical Modeling: It is useful to construct a mathematical function that represents observed data. Whether this data is linear, quadratic, rational, radical,
exponential, or logarithmic is determined by how it changes (i.e., Does the observed data change at a constant or variable rate?).
In your corporation’s Information Technology department, a database management system’s hard disk server farm is found to have the transaction bit-storage rate
characteristics listed below in Table A. Your supervisor has asked you to analyze this data to find the linear equation that best represents these rates. (Any
situation where a variable’s change with respect to a different variable’s unit change ends up always constant can be modeled by a linear function. In either Point-
Slope form,, or in Slope-Intercept form, , this constant m is the slope of the line, is the vertical intercept of the function’s graph, and the ordered pair, ,
represents some point on the line.)
Given in the table below, observations suggest that this hard disk server farm can store megabits of data in microseconds.
For each question, be sure to show all your work details for full credit. Round all value answers to three decimal places.
1. In the table below, based on the first letter of your LAST name, please select two pairs of numbers from every one of the ranges in each of the two x and y
columns. For example, if your last name starts with “M”, then you might choose and . Then, the slope of the line represented by these two points would be:
m = (y2 – y1) / (x2 – x1) = (16.2 – 15.5) / (123.5 – 121.2) = 0.7 / 2.3 = 0.304 (rounded to three decimal places). Next, the Point-Slope equation of that line is: y –
15.5 = 0.304(x – 121.2), while the Slope-Intercept form is: y = 0.304x – 21.3448.
Here, x is the independent or horizontal variable and y is the dependent or vertical variable.
If your last name does begin with M through P, do not use the same values as above.
Use your two chosen values to answer this and the following questions below.
Table A – Observations of bit-storage amounts and times.
First Letter of Your Last Name
(microseconds)
(megabits)
(microseconds)
(megabits)
A – D
105 to 106.99
12.5 to 12.99
107 to 109.99
13.0 to 13.49
E – H
110 to 111.99
13.5 to 13.99
112 to 114.99
14.0 to 14.49
I – L
115 to 116.99
14.5 to 14.99
117 to 119.99
15.0 to 15.49
M – P
120 to 121.99
15.5 to 15.99
122 to 124.99
16.0 to 16.49
Q – T
125 to 126.99
16.5 to 16.99
127 to 129.99
17.0 to 17.49
U – Z
130 to 131.99
17.5 to 17.99
132 to 134.99
18.0 to 18.49
Write your chosen two points here.
2. Use the two selected ordered pairs in part 1 to construct a linear function model in first the Point-Slope form and then convert that equation to Slope-
Intercept form for this hard disk server farm’s bit-storage equation. Show all the work details for constructing both linear function forms.
3. Graph your bit-storage function using Excel or another graphing utility. (There are free downloadable programs like Graph 4.4.2 or Microsoft’s Mathematics
4.0; or online utilities like this site ; and there are many others.) Insert the graph below these directions. Be sure to label and number the axes appropriately so
that the graph matches the chosen and calculated values from above. Also, make sure the graph contains the vertical and horizontal intercepts for your linear function.
(Bar charts, pie graphs, and similar data displays are NOT appropriate as graphs of functions in this course. Correctly constructed, your graph will be a straight
line, hence the name linear function. If you are using Excel, then make sure that you use the “Scatter with Smooth Lines and Markers” feature in the Insert tab under
the Charts menu on the Word ribbon above.)
4. Based on your model bit-storage function, how many microseconds will it take for your model to store 100 megabits? Show all the work details.
5. How many megabits can your model’s linear bit-storage function store in 200 microseconds? (“Linear forecasting” is the process of extrapolating the given
function values to an independent variable value that is not observed in the data. In this question, assuming the hard disk server farm performs at the same rate at
longer times, you are predicting or forecasting the number of megabits that can be stored in 200 microseconds without actually collecting any data at that range.) Show
all the work details.
6. As the company grows and needs faster bit-storage rates, it will expand its operations. How can this linear function model and the information it predicts with
linear forecasting help to plan for optimization and/or scaling of the future requirements for the hard disk server farm?
References
Desmos. (n.d.). Retrieved from https://www.desmos.com/
Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/
Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: https://www.microsoft.com/en-us/default.aspx
