Question

A franchise models the profit from its store as a continuous income stream with a monthly rate of flow at time t given by f(t) = 5000e0.005t (dollars per month). When a new store opens, its manager is judged against the model, with special emphasis on the second half of the first year. Find the total profit for the second 6-month period (t = 6 to t = 12). (Round your answer to the nearest dollar.)

Answer 1

To find the total profit for the second 6-month period, we integrate the given profit function f(t) = 5000e^0.005t from t=6 to t=12, evaluate the definite integral, and then round the result to the nearest dollar.

Step-by-step explanation:

To calculate the total profit for the second 6-month period, we will integrate the profit function over the appropriate interval.

Given the profit rate is modeled by f(t) = 5000e0.005t dollars per month, we want to find the total profit from

t = 6 to t = 12.

The total profit, P, over the time interval can be found by integrating the rate of profit:

P = ∫612 f(t) dt

To perform this integration, we'll follow these steps:

Integrate the function using the power rule for exponential functions.

Use the limits of integration from t = 6 to t = 12.

Evaluate the definite integral to find the total profit in the given time frame.
So, P becomes:

P = ∫612 5000e0.005t dt

P = 5000[∫ e0.005t dt] from t = 6 to t = 12

P = 5000[∫ e0.005t dt] from t = 6 to t = 12

P = (5000/0.005)e0.005t | from t = 6 to t = 12

P = (1000000)(e0.06 - e0.03)

P is then rounded to the nearest dollar.