Problem
In 1975, a wildlife resource management team introduced a certain rabbit species into a forest for the first time. In 2007 the rabbit population had grown to 7880 . The relative growth rate for this rabbit specles is $19 \%$. Use the exponential growth model $P(t)=P_{0} e^{\mathrm{tt}}$ to answer the following a) How many rabbits did the wildife resource management team introduce into the forest in 1975 ? b) How many rabbits can be expected in the year 2021 ? a) The population in 1975 was $\square$ rabbits. (Round to the nearest whole number as needed) b) The population in 2021 is expected to be $\square$ rabbits. (Round to the nearest whole number as needed)
Solution
Step 1 ：Given the exponential growth model \( P(t) = P_0 e^{rt} \)
Step 2 ：Given the population in 2007 \( P(32) = 7880 \) and the growth rate \( r = 0.19 \)
Step 3 ：Solve for the initial population \( P_0 \) using the equation \( 7880 = P_0 e^{0.19 \cdot 32} \)
Step 4 ：Calculate the population in 2021 using the initial population \( P_0 \) and the equation \( P(46) = P_0 e^{0.19 \cdot 46} \)
Step 5 ：The population in 1975 was \( \boxed{18} \) rabbits
Step 6 ：The population in 2021 is expected to be \( \boxed{112655} \) rabbits
