The rate of growth of the profit is approximated by P'(x)=xe^(-x^2) where x represents time measured in years. The total profit in the third year that the new technology is in operation is $10,000.
1.Find the total profit function
2. what happens to the total amount of profit in the long run?
To find the total profit function, we need to integrate the rate of growth of profit function.
1. To find the total profit function:
The rate of growth of profit is given by P'(x) = xe^(-x^2). To find the total profit function, we integrate P'(x) with respect to x:
∫ P'(x) dx = ∫ (xe^(-x^2)) dx
To integrate xe^(-x^2), we can use the substitution method. Let u = -x^2, then du = -2x dx. Rearranging, we have dx = -du/(2x). Substituting these values, we get:
∫ (xe^(-x^2)) dx = ∫ (e^u) (-du/2) = -(1/2) ∫ e^u du
∫ e^u du = e^u + C
Therefore, the integral becomes:
∫ (xe^(-x^2)) dx = -(1/2) ∫ e^u du = -(1/2)(e^(-x^2)) + C
Therefore, the total profit function is given by:
P(x) = -(1/2)(e^(-x^2)) + C
To find the constant C, we are given that the total profit in the third year is $10,000. We substitute x = 3 and P(x) = 10,000 into the total profit function:
10,000 = -(1/2)(e^(-3^2)) + C
Simplifying and solving for C:
10,000 = -(1/2)(e^(-9)) + C
10,000 + (1/2)(e^(-9)) = C
Therefore, the total profit function is:
P(x) = -(1/2)(e^(-x^2)) + 10,000 + (1/2)(e^(-9))
2. What happens to the total amount of profit in the long run?
In the long run, as x approaches infinity, e^(-x^2) approaches zero. This means that the profit function approaches a constant value of $10,000. In other words, the total amount of profit will stabilize and remain constant.