The R.C. Helliot Advertising Company finds that its profits on day t of an adver

Question

The R.C. Helliot Advertising Company finds that its profits on day t of an advertising campaign is given by P(t) = 3t^2 + 50t + 43,000 where P(t) is profit in dollars for day t of the campaign.

A) Find a simplified expression for the marginal profit function. (Be sure to use the proper variable in your answer.)

MP(t)=

B) What is the exact rate of change of profits when 19 days have been passed in the campaign? The rate of change is dollars per day. (Your answer may be negative.)

(C) What are their total profits on day 19 of the campaign?

Total profits on day 19 are

Explanation / Answer

solution:

profit equation is given by

P(t) = 3t^2 + 50t + 43,000

a) marginal profit function will be P'(t)

which we can get by differentiating the P(t)

P(t) = 3t^2 + 50t + 43,000

P'(t)=-6t+50

b) now rate of change of profit in 19 days = (P(19)-P(0))/(19-0)

now P(19)= -3(19)^2+50(19)+43000

   P(0)=43000

P(19)-P(0)= -3(19)^2+50(19)= -133

rate of change of profit in 19 days = (P(19)-P(0))/(19-0)= -133/19=-7

c) total profit of day19 = P(20)-P(19)

=  (-3(20)^2+50(20)+43000) -  (-3(19)^2+50(19)+43000)= -200+133 = -67

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