calculus of business

Question

. (Revenue) Tomas, the organizer of a sports event, estimates that if the event is announced 𝑥 days in advance, the revenue obtained will be 𝑅(𝑥) thousand dollars, where 𝑅(𝑥) = 400 + 120𝑥 − 𝑥 2 The cost of advertising the event for 𝑥 days is 𝐶(𝑥) where 𝐶(𝑥) = 2𝑥 2 + 300 a. Find the profit function 𝑃(𝑥) = 𝑅(𝑥) − 𝐶(𝑥), and sketch its graph. b. How many days in advance should Tomas announce the event to maximize the profit? c. What is the ratio of revenue to cost 𝑄(𝑥) = 𝑅(𝑥) 𝐶(𝑥) at the optimal announcement time found in part (b)? What happens to this ratio as 𝑥 → 0? Interpret this result

Answer 1

P(x)=370+120x-3x²

dP/dx=120-6x so that maximum occurs at x=20

lim R(x)/C(x) as x ->0 is 400/30