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.

4.3 years

Step-by-step explanation:

Given: The population of a species of starfish in the Gulf of Mexico is decreasing at an exponential rate

To find the population we have given an equation , where k is the decay rate (k<0) in t years and is the initial population.

Five years ago the population was 10,000, in 2015 it is only 2000.

So put t=5 , and A(t)= 2000 in the above function we get

Taking log on both sides, we get

Taking 2015 as the initial year  , To find t when A(t)=500

Taking log on both sides, we get

Hence, This event will occur in 4.3 years from 2015.

the answer is:

the correct option:

b.

to solve the problem, we need to remember how to operate to simplify exponential expressions using the logarithms.

we are given the equation:

then, using placing logarith with base 7 on both sides of the equation, we have:

so, rounding the obtained result to two decimal places (nearest hundreth), we have:

hence, we have that the correct option:

b.

have a nice day!