Question

A company determines that its marginal revenue per day is given by Upper R prime (t )​, where Upper R (t )is the total accumulated​ revenue, in​ dollars, on the tth day. The​ company's marginal cost per day is given by Upper C prime (t )​, where Upper C (t )is the total accumulated​ cost, in​ dollars, on the tth day.R'(t)=150e^t R(0)=0 C'(t)=150-0.1t C(0)=0

A. Find the total profit P(T) from t=0 to t=10
B. Find the average daily profit for the first ten days.

Answer 1

a) Total Profit from t = 0 to t = 10 is

$3,302,474.9

b) Average Profit over the 10 days = $330,247.5

Step-by-step explanation:

Marginal Revenue per day = R'(t) = 150eᵗ

Marginal Cost per day = C'(t) = 150 - 0.1t

where t is in days

a) Total profits over day 0 to day 10 is the definite integral of the difference between the marginal revenue per day and the marginal cost per day.

Profit = R(t) - C(t)

R = ∫¹⁰₀ [R'(t)] dt

R = ∫¹⁰₀ [150eᵗ] dt

R = [150eᵗ]¹⁰₀ = [150e¹⁰ - R(0)]

R = 3,303,969.9 - 0 = $3,303,969.9

C = ∫¹⁰₀ [C'(t)] dt

C = ∫¹⁰₀ [150 - 0.1t] dt

C = [150t - 0.05t²]¹⁰₀

C = [150(10) - 0.05(10²)] - C(0) = 1495 - 0 = $1,495

Profit = 3,303,969.9 - 1495 = $3,302,474.9

b) Average profit = (Total Profit)/(number of days)

Average profit = (3,302,474.9)/(10-0)

= $330,247.49 = $330,247.5

Hope this Helps!!!