DE Logistics growth Answers

Question 1

A population of rabbits follows a logistic growth pattern. The population starts with 18 rabbits
and the equilibrium population is 90. After 8 years there are 54 rabbits.

Find

1. The population function.

2. How long it takes to reach 85 rabbits.



Step 1: Write the logistic growth formula

𝑘𝑁0 𝑒 𝑟𝑡
𝑁(𝑡) =
𝑘 + 𝑁0 (𝑒 𝑟𝑡 − 1)


Given

𝑁0 = 18, 𝑘 = 90


Substitute these values.

90(18)𝑒 𝑟𝑡
𝑁(𝑡) =
90 + 18(𝑒 𝑟𝑡 − 1)




Step 2: Simplify

Multiply the numerator.

90(18) = 1620


So

1620𝑒 𝑟𝑡
𝑁(𝑡) =
90 + 18𝑒 𝑟𝑡 − 18
1620𝑒 𝑟𝑡
=
72 + 18𝑒 𝑟𝑡


Divide top and bottom by 18.

90𝑒 𝑟𝑡
𝑁(𝑡) =
4 + 𝑒 𝑟𝑡




Step 3: Use the information at 𝒕 = 𝟖

After 8 years,

, 𝑁(8) = 54


Substitute.

90𝑒 8𝑟
54 =
4 + 𝑒 8𝑟




Step 4: Solve for 𝒓

Multiply both sides.

54(4 + 𝑒 8𝑟 ) = 90𝑒 8𝑟


Expand.

216 + 54𝑒 8𝑟 = 90𝑒 8𝑟


Move terms.

216 = 36𝑒 8𝑟


Divide.

𝑒 8𝑟 = 6


Take logarithms.

8𝑟 = ln 6
ln 6
𝑟=
8
𝑟 = 0.22397




Step 5: Population function

Substitute into the model.

90𝑒 0.22397𝑡
𝑁(𝑡) =
4 + 𝑒 0.22397𝑡




Step 6: Find when 𝑵 = 𝟖𝟓

Substitute.