Final answer:
a. The total profit function is given by integrating the rate of growth function:
![P(t) = [1/2]te^-t^2 + C](https://img.qammunity.org/2024/formulas/business/high-school/bf8suls1h05tiv6j6uncoxo58ada91d5p2.png)
To find the constant of integration C, we use the initial condition that the total profit in the third year is $10,000:
P(3) = 10,000
Substituting t = 3 into the total profit function and solving for C:
![10,000 = [1/2]3e^-9 + CC = 10,000 - [1/2]9e^-9](https://img.qammunity.org/2024/formulas/business/high-school/u9z810jh7232yhtbga7c75e0rtg9nqtqia.png)
Therefore, the total profit function is:
![P(t) = [1/2]te^-t^2 + 10,000 - [1/2]9e^-9](https://img.qammunity.org/2024/formulas/business/high-school/bszjefz1ydibu98dzegxz172emlhyieerb.png)
b. As t gets bigger, the term te^-t^2 approaches zero more rapidly than the term e^-t^2 approaches zero. This means that in the long run, the total profit approaches the constant term:
![lim_(t- > infinity) P(t) = 10,000 - [1/2]9e^-9](https://img.qammunity.org/2024/formulas/business/high-school/9rncpv1gjoeyfi84pv5tsdtnb4pird7g72.png)
Step-by-step explanation:
The rate of growth of profit from a new technology is modeled by the function P'(t) = te^-t^2. This function represents a decreasing exponential curve that eventually approaches zero as time t increases. The initial condition that the total profit in the third year is $10,000 is used to find the constant of integration C in the total profit function P(t).
The total profit function is given by integrating the rate of growth function and solving for C using the initial condition. In the long run, as t gets bigger, the term te^-t^2 approaches zero more rapidly than the term e^-t^2 approaches zero.
This means that in the long run, the total profit approaches a constant value, which is determined by subtracting the limiting value of te^-t^2 from $10,000. This constant value represents the eventual plateau in profit that will be reached as time goes on.