Question

A rare species of insect was discovered in the Amazon Rain Forest. To protect the species, environmentalists declared the insect endangered and transplanted the insect into a protected area.The population \(P\) of the insect \(t\) months after being transplanted is $$ P(t)=\frac{50(1+0.5 t)}{2+0.01 t} $$ (a) How many insects were discovered? In other words, what was the population when \(t=0 ?\) (b) What will the population be after 5 years? (c) Determine the horizontal asymptote of \(P(t) .\) What is the largest population that the protected area can sustain?

Short answer

Initial population: 25 insects. Population after 5 years: approximately 596 insects. Largest sustainable population: 2500 insects.

Step-by-step solution

Determine Initial Population

To find the initial population of insects, substitute 0 for \(t\) in the population function \(P(t)\). Calculate \(P(0)\):\[P(0) = \frac{50(1+0.5 \times 0)}{2+0.01 \times 0} = \frac{50 \times 1}{2} = 25\]So, the initial population when \(t = 0\) is 25 insects.

Calculate Population After 5 Years

Since \(t\) is in months, 5 years is equivalent to \(5 \times 12 = 60\) months. Substitute 60 for \(t\) in the population function \(P(t)\). Calculate \(P(60)\):\[P(60) = \frac{50(1+0.5 \times 60)}{2+0.01 \times 60} = \frac{50(1+30)}{2+0.6} = \frac{50 \times 31}{2.6} = \frac{1550}{2.6} \ = 596.15\]So, the population after 5 years (or 60 months) will be approximately 596 insects.

Find Horizontal Asymptote

To determine the horizontal asymptote of \(P(t)\), evaluate the limit of \(P(t)\) as \(t\) approaches infinity. The horizontal asymptote is found by examining the leading coefficients of the highest-degree terms in the numerator and denominator.\[\lim_{{t \to \infty}} P(t) = \lim_{{t \to \infty}} \frac{50(0.5t)}{0.01t} = \frac{50 \times 0.5}{0.01} = \frac{25}{0.01} = 2500\]So, the horizontal asymptote of \(P(t)\) is 2500, which means the largest population that the protected area can sustain is 2500 insects.

Key concepts

initial population calculation

To understand how many insects were initially discovered, we start by calculating the population at the initial time, which is when the transplantation just occurred. At this time, we consider that no time has passed, so we denote it by setting the variable\( t = 0 \).
We use the provided population function \(P(t)\) to calculate this initial population: \(P(0) = \frac{50(1+0.5 \times 0)}{2+0.01 \times 0} = \frac{50 \times 1}{2} = 25 \)
Thus, the initial population of the insect species when first discovered was 25 insects. Bringing awareness to this initial population provides a baseline for understanding the population growth over time and emphasizes why the insects were declared endangered.

population prediction

To predict the insect population after a significant amount of time, let's calculate its population after 5 years. Since the function \(P(t)\) represents the population in months, we first convert 5 years into months by multiplying by 12: \(5 \times 12 = 60 \).
Now, substituting \( t = 60 \) into the population function:
\(P(60) = \frac{50(1+0.5 \times 60)}{2+0.01 \times 60} = \frac{50(1+30)}{2+0.6} = \frac{50 \times 31}{2.6} = \596.15 \)
So, the population of the insect species after 5 years (or 60 months) will be approximately 596 insects. Predicting future population sizes helps environmentalists understand how the species is recovering and adjust protection plans if necessary.

horizontal asymptote

To determine the maximum population that the protected area can sustain, we find the horizontal asymptote of the population function \(P(t)\). This requires analyzing the behavior of the function as time \(t\) approaches infinity.
Examining the leading coefficients of the highest-degree terms in both the numerator and the denominator, we have:
\(\text{lim}_{{t \to \infty }} P(t) = \text{lim}_{{t \to \infty }} \frac{50(0.5t)}{0.01t} = \frac{50 \times 0.5}{0.01} = \frac{25}{0.01} = 2500\)
Therefore, the horizontal asymptote of \(P(t)\) is 2500. This means that the largest population the protected area can sustain is 2500 insects. Calculating horizontal asymptotes is crucial as it provides insight into the long-term behavior and ultimate carrying capacity of the environment for the insect species.