Question

in 1970, a wildlife resource management team introduced a certain rabbit species into a forest for the first time. in 2002 the rabbit population had grown to 7061 the relative growth rate for this rabbit species is 23% per year. use the exponential growth model ( p(t) = p_0 e^{kt} ) to answer the following.
a. how many rabbits did the wildlife resource management team introduce into the forest in 1970?
b how many rabbits can be expected in the year 2022?
a the population in 1970 was \\( \square \\) rabbits.
(round to the nearest whole number as needed.)

Explanation:

Step1: Define variables and time difference

Let \( t=0 \) be 1970, so 2002 is \( t=2002-1970=32 \). Growth rate \( k=0.23 \), \( P(32)=7061 \).

Step2: Solve for initial population \( P_0 \)

Use \( P(t)=P_0e^{kt} \). Rearrange: \( P_0=\frac{P(32)}{e^{kt}}=\frac{7061}{e^{0.23\times32}} \). Calculate exponent: \( 0.23\times32=7.36 \). \( e^{7.36}\approx1579.5 \). So \( P_0\approx\frac{7061}{1579.5}\approx4.47 \approx4 \).

Step3: Calculate 2022 population

2022 is \( t=2022-1970=52 \). \( P(52)=4e^{0.23\times52} \). Exponent: \( 0.23\times52=11.96 \). \( e^{11.96}\approx155000 \). \( P(52)\approx4\times155000=620000 \) (rounded).

Answer:

a. 4
b. 620000