2-4 paragraphs plus two graphs
Details: Many different kinds of data can be modeled using polynomial functions.
An example of a polynomial function would be gas mileage for an automobile. If we compare gas mileage at two different speeds, V1 and V2, the gas required varies as (V1/V2), raised to the third power, (V1/V2)3.
Rational functions are also useful. For example, a cubic/cubic model can be used to explain the thermal expansion of metals with temperature. Rational functions have been used to describe problems as diverse as the movement of blood through the body to how to produce items at the lowest possible cost.
For this Discussion Board, create a set of data that can be modeled as a polynomial function. Please provide a reference to the data. Plot the data using Microsoft Excel including the equation for the fit. Discuss how closely the data seem to match to the best fit line. Do the same for data that can be modeled using a rational function. Include in your answer how this can be used in a real-life application.
To create a set of data that can be modeled as a polynomial function, we can consider the population growth of a city over time. Let's assume we have data for the population of a city at different time intervals. Here is an example set of data:
Time (years) Population
0 100
5 200
10 300
15 400
20 500
To model this data using a polynomial function, we can use Excel to create a scatter plot and then add a trendline with an appropriate degree. In this case, let's use a polynomial trendline of degree 2. The equation of the fit would be in the form of y = ax^2 + bx + c. Excel will provide us with the equation for the best fit line.
Plotting the data and adding the trendline in Excel, we can observe that the data points map relatively well to the trendline. However, it's essential to note that this is just a visual representation, and the accuracy of the fit can vary depending on the complexity of the data.
Moving on to data that can be modeled using a rational function, let us consider the voltage across a capacitor as a function of time during the process of charging. Here is an example set of data:
Time (seconds) Voltage (volts)
0 0
1 1
2 1.6
3 1.9
4 2
5 2
To model this data using a rational function, we can again use Excel to create a scatter plot and add a trendline with a rational function. For simplicity, let's consider a linear over linear rational fit. The equation of the fit would be in the form of y = (ax + b) / (cx + d).
Plotting the data and adding the rational function trendline, we can observe that the data points align quite closely with the trendline. This suggests that the rational function is a good representation of the relationship between time and voltage during the charging process of a capacitor.
In real-life applications, polynomial and rational functions can be used to model a wide range of phenomena. For example, a polynomial function can be used to predict future population growth in a city, which can aid in urban planning and resource allocation. On the other hand, a rational function can be applied to model the behavior of electrical circuits, such as the charging and discharging of capacitors, which has practical implications in electronics and power systems engineering. By fitting data to these mathematical models, we can gain insights into the underlying relationships and make informed decisions based on the predictions they provide.