Mathematical model as functions problem?

Question

a radio manufacturer can produce a particular radio at a cost of 80 dollars per radio. it is estimated that if the selling price of the radio is p dollars, then the number of radios sold per week is 180- p

a.) find the mathematical model expressing profit as function of price , p. (note: profit= revenue- cost)

b.) how many radios should be sold per week and at what price should each be sold in order to achieve maximum profit?

c.) what is the maximum profit?

Answer 1

Revenue = number of radios sold x cost of each radio

a)

R= p(180-p)

Cost =80p

Profit = P= 180p-p^2-80p

b)

P=-p^2+100p --- (1)

dP/dp=-2p+100=0

p=50, price of a radio for maximum profit

d^2P/dp^2 = -2 <0, indicates profit is maximized.

c)plug p=50 into (1)

-50^2+100(50)=-2500+5000=$2,500