Write an exponential equation from this data (data table included)

In summary, the conversation discusses finding a mathematical model for the population of a city over time using the equation y = a(b)x and the concepts of exponential growth and decay. The individual is unsure how to find the model and is seeking tips and methods for solving it. Suggestions are made to use two equations and to potentially find a best least squares fit.
  • #1
bigmac
16
0

Homework Statement


- The following table gives the population of a city over time:
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Homework Equations



I know this equation: y = a(b)x

and exponential growth: b = 1 + growth rate and b = 1 - decay rate

The Attempt at a Solution



I know from back in chapter 2 that first differences = linear model, second differences are the same = quadratic and if the 3rd differences are the same then its a cubic model...but that doesn't work here. I am completely stuck...how do I find the model. Any hints/tips/methods will be greatly appreciated! Thanks!
 
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  • #2
There are two unknown constants in your formula, a and b. You need two equations to solve for two unknowns so pick two points (typically, it is best to endpoints, here 1954 and 1994).

That will give you [tex]132459= ab^{1954}[/tex] and [tex]514013= ab^{1994}[/tex]

It should be easy to see that dividing one equation by the other will eliminate a, leaving a single equation to solve for b.
 
  • #3
Perhaps you are supposed to find a best least squares fit to the data?
 

What is an exponential equation?

An exponential equation is a mathematical equation in which the variable appears in the exponent. It can be written in the form of y = ab^x, where a is the initial value, b is the growth factor, and x is the independent variable.

How do I know if the given data can be represented by an exponential equation?

If the data shows a constant growth or decay rate, it can be represented by an exponential equation. This means that the ratio of any two consecutive data points remains the same.

Can you provide an example of creating an exponential equation from a data table?

Sure, let's say we have the following data table:

x y
0 1
1 3
2 9
3 27

To create an exponential equation, we can use the formula y = ab^x and plug in the values from the table. We can start by choosing any two data points, for example (0, 1) and (1, 3). This gives us the equation 1 = ab^0 and 3 = ab^1. Solving for a and b, we get a = 1 and b = 3. Therefore, the exponential equation for this data is y = 3^x.

Can an exponential equation be used to make predictions?

Yes, an exponential equation can be used to make predictions about future values. It is based on the assumption that the growth or decay rate will remain constant. However, it is important to note that predictions may not always be accurate as real-life situations can be affected by various other factors.

Are there any limitations to using an exponential equation for data analysis?

While exponential equations can be useful in representing data and making predictions, they may not be suitable for all types of data. Some situations may involve non-constant growth or decay, in which case other types of equations may be more appropriate. It is important to carefully analyze the data and consider other factors before using an exponential equation for data analysis.

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