Question

12 mark for review
$f(x) = 4,000(0.90)^x$
a conservation scientist implemented a program to reduce a certain invasive insect population in an area. the given function estimates this insect species population $x$ years after 2008, where $x \leq 8$. which of the following is the best interpretation of 4,000 in this context?
a the estimated percent decrease in the insect population for this species and area every 8 years after 2008.
b the estimated initial insect population for this species and area in 2008
c the estimated percent decrease in the insect population for this species and area each year after 2008
d the estimated insect population for this species and area 8 years after 2008

Explanation:

The function is of the form \( f(x)=a(b)^x \), where \( a \) is the initial value, \( b \) is the growth/decay factor, and \( x \) is the time. Here, \( x \) represents years after 2008. When \( x = 0 \) (which is the year 2008), \( f(0)=4000(0.90)^0=4000\times1 = 4000 \). So 4000 is the initial population in 2008. Option A is incorrect as it refers to percent decrease over 8 years, but 4000 is a population, not a percent. Option C is incorrect as the percent decrease is related to \( 1 - 0.90=0.10 \) (10% decrease per year), not 4000. Option D: For \( x = 8 \), \( f(8)=4000(0.90)^8
eq4000 \), so it's not the population 8 years after.

Answer:

B. The estimated initial insect population for this species and area in 2008