the function p models the population of rabbits on a farm and is given by ( p(t)=\frac{200}{1 + 5t} ) for (…

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?\na the population decreases and approaches a value of 0 rabbits.\nb the population increases and approaches a value of 40 rabbits.\nc the population increases and approaches a value of 200 rabbits.\nd the rabbit population increases without bound.

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?\na the population decreases and approaches a value of 0 rabbits.\nb the population increases and approaches a value of 40 rabbits.\nc the population increases and approaches a value of 200 rabbits.\nd the rabbit population increases without bound.

Answer

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\rightarrow\infty}P(t)=\lim_{t\rightarrow\infty}\frac{200}{1 + 5t}). Divide numerator and denominator by (t): (\lim_{t\rightarrow\infty}\frac{\frac{200}{t}}{\frac{1}{t}+5}). Since (\lim_{t\rightarrow\infty}\frac{1}{t}=0) and (\lim_{t\rightarrow\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.