Question

An endangered species of fish has a population that is decreasing exponentially

The population 8 years ago was 1900. Today, only 1100 of the fish are alive. Once the population drops below 100, the situation will be irreversible. When will this happen, according to the model? (Round to the nearest whole year.)

Answer 1

For exponential decreasing, the formula is P(t)=P(0)e^-kt, where P(0)is the initial amount, k is the constant of decrease, and t is time. So:
1100=1900*e^-8k
0.57894736842105263157894736842105=e^-8k
ln 0.57894736842105263157894736842105=ln e^-8k=-8k ln e=-8k
k=0.06831796329600873949338548174034
So, the population reaches 100 in:
100/1900=e^-0.06831796329600873949338548174034t
0.05263157894736842105263157894737=e^-0.06831796329600873949338548174034t
ln 0.05263157894736842105263157894737=ln e^-0.06831796329600873949338548174034t=
-0.06831796329600873949338548174034t ln e=-0.06831796329600873949338548174034t
t=43.0990 years
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