Question

the function p models the population of rabbits on a farm and is given by ( p(t)=\frac{200}{1 + 5t} ) for ( tgeq0 ), where t is measured in months since the start of the year. which of the following describes the population of the rabbits as time increases?
a the population decreases and approaches a value of 0 rabbits.
b the population increases and approaches a value of 40 rabbits.
c the population increases and approaches a value of 200 rabbits.
d the rabbit population increases without bound.

Explanation:

Step1: Analyze the function as \(t\) increases

We have \(P(t)=\frac{200}{1 + 5t}\). As \(t\) (where \(t\geq0\)) increases, the denominator \(1+5t\) increases.

Step2: Find the limit as \(t\to\infty\)

\(\lim_{t
ightarrow\infty}P(t)=\lim_{t
ightarrow\infty}\frac{200}{1 + 5t}\). Divide numerator and denominator by \(t\): \(\lim_{t
ightarrow\infty}\frac{\frac{200}{t}}{\frac{1}{t}+5}\).
Since \(\lim_{t
ightarrow\infty}\frac{1}{t}=0\) and \(\lim_{t
ightarrow\infty}\frac{200}{t}=0\), we get \(\frac{0}{0 + 5}=0\). Also, when \(t = 0\), \(P(0)=\frac{200}{1+5\times0}=200\). As \(t\) increases from \(0\) to \(\infty\), \(P(t)\) is a decreasing function.

Answer:

A. The population decreases and approaches a value of 0 rabbits.