Question

14 the population of african elephants has been decreasing at an alarming rate for the past century. in the late 1800s, there were several million african elephants. in the 1990s, that number dropped to only about half a million. the goal of conservationists is to collect and analyze scientific data to work toward a better future for the species. the population decline of african elephants can be modeled by an exponential function ( f(x) = a(b)^x ), where ( f(x) ) is the elephant population in millions and ( x ) is the number of years since 1990. if the elephant population was 12 million in 1900 and declined at 7.5% per year, write the function ( f(x) ) that relates the elephant population to the number of years since 1990. if the population decline continues, estimate the population in 2020. explain

Explanation:

Step1: Define known variables

Let \(P(t)\) = elephant population (millions), \(t\) = years since 1900.
Given: \(P(0) = 12\), decay rate \(r = 0.075\) (7.5% = 0.075)

Step2: Set up decay function

Exponential decay formula: \(P(t) = P_0(1-r)^t\)
Substitute values: \(P(t) = 12(1-0.075)^t = 12(0.925)^t\)

Step3: Calculate \(t\) for 2020

\(t = 2020 - 1900 = 120\)

Step4: Compute 2020 population

Substitute \(t=120\):
\(P(120) = 12(0.925)^{120}\)
First calculate \(0.925^{120} \approx 1.017 \times 10^{-4}\)
Then \(P(120) \approx 12 \times 1.017 \times 10^{-4} \approx 0.00122\)

Answer:

The estimated African elephant population in 2020 is approximately 0.00122 million, or 1,220 individuals.