Exponential Equations

Equations

The equation problems may seem a bit more difficult, but once you understand the growth and decay formula, they become simple and straightforward.

For example:

A field of trees is being attacked by a poisonous fungus.  Before the fungus arrived, there were 460 trees in the field.  The population is reducing at a rate of /(17%/) each year.  Which of the following functions, N, models the number of trees in the field t years later?

A) \(N(t)=.17(460)t\)
B) \(N(t)=460\left(0.17\right)^t\)
C) \(N(t)=460(.83)^t\)
D) \(N(t)=460\left(1.17\right)^t\)

Strategy:

  1. Plug values into the exponential growth or decay formula.
  2. Solve.

Exponential growth formula:

$$I\left(1+r\right)^t$$

Exponential decay formula:

$$I\left(1-r\right)^t$$

\(I=\) the initial amount.

\(r=\) the rate of change as a decimal (e.g. \(25%=.25\)).

\(t=\) time elapsed.

The initial amount and the rate of change will usually be easy to find.  The only exceptions are when the SAT says something “doubles,” or “halves.” 

Doubles means it is exponential growth and \(r=1\).  

Halves means it is exponential decay and \(r=.5\).

There you have it! Plug numbers into these two simple formulas and these problems become easy.

Back to our example:

The problem discusses a population of trees that is reducing: this means we use the exponential decay formula.  The initial amount is 460 trees, and the rate of change is \(17%\) as a decimal-\(.17\) .

$$460\left(1-.17\right)^t=460\left(.83\right)^t$$

The correct answer is C.

Sometimes, you may need to apply this equation to information from tabular data as well!