Exponential (Population) Growth Cont.
exponential growth
If a quantity starts at size [latex]P_0[/latex] and grows by [latex]R\%[/latex] (written as a decimal, [latex]r[/latex]) every time period, then the quantity after [latex]n[/latex] time periods can be determined using either of these relations:
Recursive form
[latex]P_n = (1+r)P_{n-1}[/latex]
Explicit form
[latex]P_n = (1+r)^{n}P_0[/latex] or equivalently, [latex]P_n= P_0(1+r)^{n}[/latex]
We call [latex]r[/latex] the growth rate.
The term [latex](1+r)[/latex] is called the growth multiplier, or common ratio.
To evaluate expressions like [latex](1.03)^{6}[/latex], it will be easier to use a calculator than multiply [latex]1.03[/latex] by itself six times. Most scientific calculators have a button for exponents. It is typically either labeled like:
^, [latex]y^{x}[/latex] , or [latex]x^{y}[/latex].
To evaluate [latex]1.03^{6}[/latex] we’d type [latex]1.03 ^ 6[/latex], or [latex]1.03 y^{x} 6[/latex]. Try it out – you should get an answer around [latex]1.1940523[/latex].
As a note on rounding, notice that if we had rounded the growth rate to [latex]2.1\%[/latex], our calculation for the emissions in 2050 would have been [latex]3347[/latex]. Rounding to [latex]2\%[/latex] would have changed our result to [latex]3156[/latex]. A very small difference in the growth rates gets magnified greatly in exponential growth. For this reason, it is recommended to round the growth rate as little as possible.
If you need to round, keep at least three significant digits – numbers after any leading zeros. So [latex]0.4162[/latex] could be reasonably rounded to [latex]0.416[/latex]. A growth rate of [latex]0.001027[/latex] could be reasonably rounded to [latex]0.00103[/latex].
In the previous example, we had to calculate the [latex]10th[/latex] root of a number. This is different than taking the basic square root, √. Many scientific calculators have a button for general roots. It is typically labeled like:
[latex]\sqrt[y]{x}[/latex]
To evaluate the [latex]3rd[/latex] root of [latex]8[/latex], for example, we’d either type [latex]3[/latex] [latex]\sqrt[x]{{}}[/latex][latex]8[/latex], or [latex]8[/latex][latex]\sqrt[x]{{}}[/latex][latex]3[/latex], depending on the calculator. Try it on yours to see which to use – you should get an answer of [latex]2[/latex].
If your calculator does not have a general root button, all is not lost. You can instead use the property of exponents which states that:
[latex]\sqrt[n]{a}={a}^{\frac{1}{2}}[/latex].
So, to compute the [latex]3rd[/latex] root of [latex]8[/latex], you could use your calculator’s exponent key to evaluate [latex]8^{\frac{1}{3}}[/latex]. To do this, type:
[latex]8 y^{x} ( 1 ÷ 3 )[/latex]
The parentheses tell the calculator to divide [latex]\frac{1}{3}[/latex] before doing the exponent.
- Find more than two pieces of data. Plot the values, and look for a trend. Does the data appear to be changing like a line, or do the values appear to be curving upwards?
- Consider the factors contributing to the data. Are they things you would expect to change linearly or exponentially? For example, in the case of carbon emissions, we could expect that, absent other factors, they would be tied closely to population values, which tend to change exponentially.