The population of a species of starfish in the Gulf of Mexico is decreasing at an exponential rate, A (t) = A0e (kt). Five years ago the population was 10,000, in 2015 it is only 2000. When the population is 500, the starfish population cannot recover. When will this event occur?

Show your work and explain the process of solving this problem.

First is to solve the value of k

when A0 = 10000

a (t) = 2000

t = 5

using the formula

a (t) = A0 e^ (kt)

2000 = 10000 e^ (5k)

2000 / 10000 = e^ (5k)

ln (1/5) = 5k

k = (ln (1/5)) / 5

k = - 1.6094 / 5

k = - 0.3219

now solve for t,

500 = 10000 e^ (-0.3219t)

ln (500/10000) = - 0.3219t

t = 9.3067 years

since that is calculated as the basis at 5 years ago

then actual t is 4.3067 years