Question

the exponential model below represents the population of a animal $x$ years from now. what is meaning of 1.08 in the model?
$f(x)=1861(1.08)^x$
show your work here
hint: to add an exponent ($x^y$), type \exponent\ or press \^\
$circ$ every year, the animal decreases in population by 8%
$circ$ the animals population will be zero after 8 years
$circ$ every year, the animal increases in population by 108%
$circ$ every year, the animal decreases in population by 108%
$circ$ every year, the animal increases in population by 8%
$circ$ the animal will have doubled in population after 8 years

Explanation:

Step1: Recall exponential growth form

The standard exponential growth model is $f(x) = a(1+r)^x$, where $a$ is the initial amount, $r$ is the annual growth rate, and $1+r$ is the growth factor.

Step2: Match given model to standard form

Given $f(x) = 1861(1.08)^x$, we identify $1+r = 1.08$.

Step3: Calculate growth rate $r$

$r = 1.08 - 1 = 0.08$, which is $8\%$. This means the population grows by 8% annually.

Step4: Eliminate incorrect options

• Options about decrease are wrong (1.08 > 1, so it's growth).
• 108% increase would mean $1+r=2.08$, not 1.08.
• The model does not predict a zero population or doubling in 8 years.

Answer:

Every year, the animal increases in population by 8%