An endangered species of fish has a population that is decreasing exponentially:A=A0 e^kt The population 10 years ago was 1700. Today, only 800 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.) A. 29B. 30C. 32D. 31E. None of the above

The correct answer is E. none of the above. The population will drops below 100 when t ≥ 38.

Explanation:

Given A = A0 e^kt. The population 10 years ago is A0, the population today is A (10), and we have to find the value of "k" and then the time when population drops below 100.

So, A (t) = 1700 e^kt ⇒ A (10) = 1700 e^k (10) ⇒ 800 = 1700 e^k (10) ⇒

800/1700 = e^k (10) ⇒ln (800/1700) = k (10) ln e ⇒ - 0.754/10 = k ⇒

k = - 0.0754.

Now you have all the parameters, so you can find the time at which the population drops below 100.

A (t) = 1700 e^kt ⇒ 100 = 1700 e^ (-0.0754) t ⇒100/1700 = e^ (-0.0754) t ⇒

ln (100/1700) = (-0.0754) t ln e ⇒ [ln (100/1700) ] / (-0.0754) = t ⇒

t = 38.

So, the population will drops below 100 when t ≥ 38.