A few days ago

Help!!!! Maximum Profit problem?

A company begins an advertising campaign in a certain city to market a new product. The percentage of the target market that buys the product is a function of the length of the advertising campaign. The company estimates this percentage as 1-e^-0.05t where t=number of days of the campaign. The target market is estimated to be 1,000,000 people and the price per unit is $0.40. The cost of advertasing is $1000 per day.

Find the length of the advertising campaign that will result in the maximum profit.

P.S. Please list all the steps taken to answer the problem. Im interested in learning also, not just getting the answer.

Top 1 Answers
A few days ago

Favorite Answer

So profit = revenue – cost. You didn’t say what the cost of making each unit is, so it must be 0 which doesn’t really make sense but if advertising is the only cost,

= (1 – e^ – 0.05t)(1,000,000)(0.40) – 1000t

= (1 – e^ -0.05t)(400,000) – 1000t

its derivative is 400,000(0.05e^ -0.05t) – 1000 = 0 to be a max

(400,000)0.05 e^ -0.05t = 1000

20,000e^-0.05t = 1000

e^-0.05t = 1000/20000 = 0.05

-0.05t ln (e) = ln(0.05)

0.05t = 2.9957…, divided by 0.05 = 59.9 days

Take this for what it costs: I’m not all that confident in it