) A company determines that its marginal revenue per day is given by R'(t) = 100et , 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) = 140 - 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 = 5 (the first 5 days). Round to the nearest dollar. Note: P(T) = R(T) - C(T) = T 0 ∫ [R'(t) - C'(t)] dt.

$14038

Explanation:

The company has marginal revenue R'(t) = . Therefore its revenue R(t) is given as;

R(t) = ∫R'(t)

R(t)= ∫ dt =   + c

R(t) =   + c

But R(0) = 0, therefore:

R(0) =   + c = 0

+ c = 0

100 + c =0

c = -100

Also the marginal cost per day is given by C'(t) = 140 - 0.3t

C'(t) = 140 - 0.3t

C(t) = ∫C(t) = ∫ (140 - 0.3t) dt = 140t - (0.3/2) t² + C

But C(0) = 0

C(0) = 140 (0) - (0.3/2)(0)² + c = 0

c = 0

C(0) = 140t - (0.3/2) t²

Profit P(t) = R(T) - C(T) , hence the total profit from t = 0 to t = 5 is given as:

P(t) =

The profit is $14038

reducing the scale of the market for a new product.