Question

A company determines that its marginal revenue per day is given by R'(t) = 80e^t with superscript (t), R(0) = 0, where R(t) = the revenue, in dollars, on the tth day. The company's marginal cost per day is given by C'(t) = 150 - 0.3t, C(0) = 0, where C(t) = the cost, in dollars, on the tth day. Find the total profit from t = 0 to t = 6 (the first 6 days). Round to the nearest dollar.

Answer 1

To find the total profit from t = 0 to t = 6, calculate the revenue and cost using the given functions and subtract the cost from the revenue.

Step-by-step explanation:

To find the total profit from t = 0 to t = 6, we need to calculate the revenue and cost from t = 0 to t = 6 and then subtract the cost from the revenue. First, let's find the revenue. The revenue function is given as R(t) = ∫(0 to t) R'(u) du = ∫(0 to t) 80e^u du. Integrating this function gives R(t) = 80e^t - 80. Now let's find the cost. The cost function is given as C(t) = ∫(0 to t) C'(u) du = ∫(0 to t) (150 - 0.3u) du. Integrating this function gives C(t) = 150t - 0.15t^2. Finally, we can find the total profit by subtracting the cost from the revenue: P(t) = R(t) - C(t) = (80e^t - 80) - (150t - 0.15t^2). To find the total profit from t = 0 to t = 6, we can evaluate P(t) at t = 6: P(6) = (80e^6 - 80) - (150(6) - 0.15(6)^2). Round the result to the nearest dollar to find the total profit.