An auto dealer offers an additional discount to fleet buyers who purchase one or

Question

An auto dealer offers an additional discount to fleet buyers who purchase one or more new Cadillac Sevilles. To encourage sales, the dealer reduces the price of each car by a percentage equal to the total number of cars purchased. For example, a fleet buyer purchasing 15 cars will receive a 15% discount. a. Assuming that the pre-incentive price of a Cadillac Seville is $41,000, write a model for the after-incentive price of each car as a function of the number of cars purchased. b. Write a model for the auto dealer's revenue as a function of the number of cars purchased by the fleet buyer. c. How many cars should the dealer sell in order to maximize revenue? Show some work (i.e. calculus) to support your answer.

Explanation / Answer

12)

a)let number of car purchased = x,after-incentive price per each car = y

pre-incentive price per each car = 41000

after-incentive price per each car, y = 41000*(1-(x/100))

b)

revenue ,R(x,y)=x*y

revenue ,R(x)=x* 41000*(1-(x/100))

revenue ,R(x)=41000(x-(x2/100))

c)

for maximum revenue, dR/dx =0 ,d2R/dx2<0

dR/dx =41000(1-(x/50)) ,d2R/dx2=41000(0-(1/50))

41000(1-(x/50))=0 ,d2R/dx2=-820<0

=>x/50 =1

=>x=50

checking end points:

when 0 cars are sold , revenue =0

when 100 cars are sold , revenue =0

so dealer should sell 50 cars in order to maximise the revenue

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