For G(t) = 10,000/(1+24e^-1.2t) when does the population start to decrease?
I know this question is looking for the maxima for the function, but I can't quite get there. Could someone show me the answer with the steps?
First, we will find the initial population, when t= 0
==> G(0)= 10,000/(1+24)= 10,000/25= 400
Now we need to find t such that G(t) < 400
10000/(1+24e^(-1.2t)) < 400
Use reciprocal and reverse the inequality:
==> (1+24e^(-1.2t))/10,000 > 1/400
Multiply by 10,000
==> 1+24e^(-1.2t) > 25
==> 24e^(-1.2t) > 24
==> e^(-1.2t) > 1
==> -1.2t > ln 1
==> -1.2t > 0
==> t < 0
t< 0 that means the population drcreases when t is a negative number. But time can not be negative.
Then, for all values where t>0 , the population increases.
Then, we notice that the population does not decrease
To find when a function decreases, find when the derivative is negative.
`G'(t) = ((10000)(-1))/(1+24e^(-1.2t))^2 (24)(-1.2)e^(-1.2t) = (288,000e^(-1.2t))/(1+24e^(-1.2t))`
This is always positive, so the function never decreases, G(t) gets closer and closer to 10,000 but never reaches it. You had trouble with finding the maximum because it's maximum is at t=oo. The minimum is at t=0 G(0) = 10,000/25 = 400.