Skill: 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.