Question

2. The rate of growth of the profit (in millions of dollars) from a new technology is approximated by P′ (t)=te−t^2 where t represents time measured in years. The total profit in the third year that the new technology is in operation is $10,000. a. Find the total profit function. b. What happens to the total profit in the long run, as t gets bigger? Write in a complete sentence.

Answer 1

To find the total profit function, we integrate the rate of growth of profit function with respect to time. The total profit function is P(t) = -0.5e^(-t^2) + 10000 + 0.5e^(-9). In the long run, as t gets bigger, the total profit will approach a constant value of $10,000.

Step-by-step explanation:

To find the total profit function, we need to integrate the rate of growth of profit function with respect to time. The rate of growth of profit function is given as P'(t) = te^(-t^2). So, integrating it with respect to t gives us the total profit function P(t).

Integrating P'(t) = te^(-t^2) gives us P(t) = -0.5e^(-t^2) + C. Now, we can find the value of C using the given information that the total profit in the third year is $10,000.

Substituting t = 3 and P(t) = 10000 into the total profit function, we get 10000 = -0.5e^(-9) + C. Solving for C gives us C = 10000 + 0.5e^(-9). So, the total profit function is P(t) = -0.5e^(-t^2) + 10000 + 0.5e^(-9).

As t gets bigger (approaches infinity), the term e^(-t^2) approaches 0. So, in the long run, the total profit will approach a constant value of $10,000.

Answer 2

a. The total profit function is given by integrating the rate of growth function:

`P(t) = [1/2]te^-t^2 + C`

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`

Therefore, the total profit function is:

`P(t) = [1/2]te^-t^2 + 10,000 - [1/2]9e^-9`

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`

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.