Expert: Paul Klarreich - 12/10/2009

Question
Fruit flies are placed in a half pink milk bottle with a banana(for food) and yeast plants(for food and to provide a stimulus to  to lay eggs). Suppose that the fruit fly population after t days is given by    P(t)= 230/1+56.5^-0.37t

determine the initial population
determine the maximum population. Justify/explain your answer
what is the population after 5 days?
and how long does it take for the population to reach 180? Give exact answer and evaluate exact answer to 2 decimal places


Thank you so much and I am sorry that I am asking so many questions for the same problem.

Thank you again,
Delmy


Answer
Questioner: Delmy
Country: United States
Category: Advanced Math
Private: No
Subject: precal
Question: Fruit flies are placed in a half pink milk bottle with a banana(for food) and yeast plants(for food and to provide a stimulus to  to lay eggs). Suppose that the fruit fly population after t days is given by    P(t)= 230/1+56.5^-0.37t

determine the initial population
determine the maximum population. Justify/explain your answer
what is the population after 5 days?
and how long does it take for the population to reach 180? Give exact answer and evaluate exact answer to 2 decimal places


Thank you so much and I am sorry that I am asking so many questions for the same problem.

Thank you again,
Delmy

...........................................
I assume you mean:

P(t)= 230/( 1+56.5^-0.37t)

otherwise your question makes no sense.

1. This is P(0) = 230/2

2. As t -> infinity,  56.5^-0.37t --> 0.  Take it from there.

3. This is  P(5).  Use your calculator.

4. Solve:  180 = 230/1+56.5^-0.37t

Not so easy:

180 = 230/1+56.5^-0.37t

1+56.5^-0.37t = 230/180

56.5^-0.37t = 230/180 - 1

(e^(ln 56.5))^-0.37t = 23/18 - 1

e^-0.37(ln 56.5)t = 23/18 - 1

-0.37(ln 56.5)t = ln(23/18 - 1)

t = ln(23/18 - 1)/(-0.37(ln 56.5))

That's a little calculator work, but worth it.