An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Determine the constant A using the initial condition P ( 0 ) = 500 , resulting in A = 2 .
Determine the constant k using the condition P ( 3 ) = 800 , resulting in k ≈ 0.2756 .
Substitute the values of K , A , and k into the logistic function P ( t ) = 1 + A e − k t K .
The final logistic function is P ( t ) = 1 + 2 e − 0.2756 t 1500 .
P ( t ) = 1 + 2 e − 0.2756 t 1500
Explanation
Problem Analysis We are given the initial population, carrying capacity, and population after 3 years. We need to find the logistic function that models this population growth.
Logistic Function The logistic function is given by the formula: P ( t ) = 1 + A e − k t K where:
P ( t ) is the population at time t
K is the carrying capacity
A is a constant determined by the initial population
k is the growth rate
Finding A We know that the initial population P ( 0 ) = 500 and the carrying capacity K = 1500 . We can use the initial population to find the constant A :
P ( 0 ) = 1 + A e − k ( 0 ) K 500 = 1 + A e 0 1500 500 = 1 + A 1500 1 + A = 500 1500 1 + A = 3 A = 2
Finding k Now we know that after 3 years, the population is 800, so P ( 3 ) = 800 . We can use this information to find k :
P ( 3 ) = 1 + A e − 3 k K 800 = 1 + 2 e − 3 k 1500 1 + 2 e − 3 k = 800 1500 1 + 2 e − 3 k = 1.875 2 e − 3 k = 0.875 e − 3 k = 0.4375 − 3 k = ln ( 0.4375 ) k = − 3 ln ( 0.4375 ) k ≈ 0.2756
Final Logistic Function Now we have all the constants, so we can write the logistic function: P ( t ) = 1 + 2 e − 0.2756 t 1500 Therefore, the logistic function that models this population as a function of time is: P ( t ) = 1 + 2 e − 0.2756 t 1500
Final Answer The logistic function that models the population growth is: P ( t ) = 1 + 2 e − 0.2756 t 1500
Examples
Logistic functions are incredibly useful in various real-world scenarios. For instance, they can model the spread of a disease through a population, the growth of a company's sales after a new product launch, or even the adoption rate of a new technology. Understanding logistic growth helps in making informed decisions about resource allocation, marketing strategies, and public health interventions. By analyzing the initial growth phase, the saturation point, and the factors influencing these stages, we can better predict and manage real-world phenomena.
