Ten blackberry plants started growing in my yard. Absent constraint, blackberries
will spread by 200% a month. My yard can only sustain about 50 plants. Using the
logistic growth model,
a. Write a recursive formula for the number of blackberry plants in my yard
b. Calculate the number of plants after 1, 2, and 3 months
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To solve this problem using the logistic growth model, we need to consider the capacity constraint of your yard, which can only sustain about 50 plants. The logistic growth model takes into account this constraint and provides a way to predict the population growth over time.
a. Recursive formula for the number of blackberry plants in your yard:
To write a recursive formula, we need to establish a relationship between the current population and the population in the previous month.
Let P(n) be the number of blackberry plants in your yard after n months. We can define the recursive formula as follows:
P(n) = P(n-1) + 2 * P(n-1) * (1 - P(n-1)/50)
This formula represents the population after n months as the sum of the previous month's population plus the growth rate. The growth rate is calculated by multiplying the previous month's population by 2 and then adjusting it based on the difference between the previous population and the capacity of the yard (50 plants). The term (1 - P(n-1)/50) represents the proportion of the yard's capacity that is still available for growth.
b. Calculation of the number of plants after 1, 2, and 3 months:
To calculate the number of plants after a given number of months, we can use the recursive formula:
After 1 month:
P(1) = P(0) + 2 * P(0) * (1 - P(0)/50)
Substituting P(0) = 10 (initial number of plants):
P(1) = 10 + 2 * 10 * (1 - 10/50)
= 10 + 2 * 10 * (1 - 0.2)
= 10 + 2 * 10 * 0.8
= 10 + 16
= 26
After 2 months:
P(2) = P(1) + 2 * P(1) * (1 - P(1)/50)
Substituting P(1) = 26:
P(2) = 26 + 2 * 26 * (1 - 26/50)
= 26 + 2 * 26 * (1 - 0.52)
= 26 + 2 * 26 * 0.48
= 26 + 24.96
= 50.96 (rounded to 2 decimal places)
After 3 months:
P(3) = P(2) + 2 * P(2) * (1 - P(2)/50)
Substituting P(2) = 50.96:
P(3) = 50.96 + 2 * 50.96 * (1 - 50.96/50)
= 50.96 + 2 * 50.96 * (1 - 1.0192)
= 50.96 + 2 * 50.96 * (-0.0192)
= 50.96 + 2 * -0.3916
= 50.96 - 0.7832
= 50.1768 (rounded to 2 decimal places)
Therefore, after 1 month, the number of plants is 26, after 2 months it is approximately 50.96, and after 3 months it is approximately 50.18.
Ten blackberry plants started growing in my yard. Absent constraint, blackberries
will spread by 200% a month. My yard can only sustain about 50 plants. Using the
logistic growth model,
a. Write a recursive formula for the number of blackberry plants in my yard
b. Calculate the number of plants after 1, 2, and 3 months