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Advanced Math / Nonlinear functions Difficulty: Medium

$f left parenthesis x right parenthesis equals 4,000 left parenthesis 0.75 right parenthesis Superscript x$

An entomologist recommended a program to reduce a certain invasive beetle population in an area. The given function estimates this beetle species' population $x$ years after $2012$ , where $x less than or equals 7$ . Which of the following is the best interpretation of $4,000$ in this context?

The estimated initial beetle population for this species and area in $2012$

The estimated beetle population for this species and area $7$ years after $2012$

The estimated percent decrease in the beetle population for this species and area each year after $2012$

The estimated percent decrease in the beetle population for this species and area every $7$ years after $2012$

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Explanation

Choice A is correct. For an exponential function in the form $f (x) = a {(b)}^{x}$ , where $a$ and $b$ are positive constants and $b less than 1$ , the initial value of $f (x)$ , or the value of $f (x)$ when $x equals 0$ , is $a$ and the value of $f (x)$ decreases by $100 left parenthesis 1 minus b right parenthesis percent sign$ each time $x$ increases by $1$ . Therefore, the initial value of the function $f left parenthesis x right parenthesis equals 4,000 left parenthesis 0.75 right parenthesis Superscript x$ , or the value of $f (x)$ when $x equals 0$ , is $4,000$ . Therefore, the best interpretation of $4,000$ in this context is the estimated initial beetle population for this species and area in $2012$ .

Choice B is incorrect. The estimated beetle population for this species and area $7$ years after $2012$ is $4,000 left parenthesis 0.75 right parenthesis Superscript 7$ , or approximately $534$ , not $4,000$ .

Choice C is incorrect. The estimated percent decrease in the beetle population for this species and area each year after $2012$ is $100 left parenthesis 1 minus 0.75 right parenthesis$ , or $25$ , not $4,000$ .

Choice D is incorrect. The estimated percent decrease in the beetle population for this species and area every $7$ years after $2012$ is $100 left parenthesis 1 minus 0.75 Superscript 7 Baseline right parenthesis$ , or approximately $87$ , not $4,000$ .