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**:

**Plug values into the exponential growth or decay formula.****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!

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