The bat equation

Question

A species of bat is in danger of becoming extinct. Five years ago, the total population of the

species was 2000. Two years ago, the total population of the species was 1350. What was the

total population of the species one year ago? n = n₀e^kt

(hint: what would I call year 1)

(show work and round to tenths.)

Answer 1

Utilizing the equation n=n₀e^kt, we can take an initial time of t=0 5 years ago. Then, we know that the exponent is zero, so the population is n=n₀=2000, so n₀=2000. Then, to find the value of k, we know that the population at time t=3 was 1350, so 1350=2000*e^3k, so 1350/2000=e^3k, so k=ln(.675)/3. Therefore, to find the total population 1 year ago, when t=4, we plug into the equation we just solved for, so n = 2000* e^kt. So n=2000*e^{4/3 * ln(.675)}, which we can simplify utilizing exponent power rules, to 2000*.675^4/3=1184.2