Model exponential growth and decay

Exponential growth and decay models represent quantities that change by a constant multiplicative factor over equal intervals, using forms such as $A(t)=A_0b^t$ or $A(t)=A_0(1+r)^t$ . The initial value, growth or decay factor, and percentage rate are interpreted from equations, tables, graphs, and contexts, with $b>1$ indicating growth and $0<b<1$ decay; this distinguishes exponential change from constant additive change and excludes differential-equation-based or more abstract generalizations.

Detailed Explanation: Model exponential growth and decay

An exponential model is used when a quantity changes by the same multiplicative factor over equal time intervals. Its general form is

A(t)=A_0b^t,

where:

$A_0$ is the initial amount,
$b$ is the growth or decay factor,
$t$ is the number of time intervals.

If the quantity grows by $r$ percent each interval, then

b=1+r.

If it decays by $r$ percent each interval, then

b=1-r.

A value of $b>1$ represents growth, while $0<b<1$ represents decay.

Example

A population of bacteria starts at $500$ . It increases by $12\%$ every hour. Model the population after $t$ hours and find the population after $6$ hours.

Step 1: Identify the initial amount.

The starting population is

A_0=500.

Step 2: Convert the percent increase to a growth factor.

A $12\%$ increase means multiply by

1+0.12=1.12.

Thus, the growth factor is $b=1.12$ .

Step 3: Write the exponential model.

Substitute the initial amount and growth factor into $A(t)=A_0b^t$ :

A(t)=500(1.12)^t.

This model shows that the population is multiplied by $1.12$ every hour.

Step 4: Find the population after $6$ hours.

Substitute $t=6$ :

A(6)=500(1.12)^6.

A(6)\approx 986.91.

Since a population is counted in whole bacteria, the population is approximately

\boxed{987\text{ bacteria}}.

This is exponential growth because the population increases by a constant percentage, not by a constant number. A model such as $500+12t$ would represent adding $12$ bacteria each hour, which is linear rather than exponential.

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Exponential Function Solving - Growth (Discrete) Scenario to Value at Time

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