1) A telephone company has 10,000 phones in a certain city where the basic rate is $4 per month. The officials have evidence that if the charge is reduced the number of phones will increase at an estimated rate of 1,000 phones for each 25 cent reduction. What rate would yield the greatest income?

Question

1) A telephone company has 10,000 phones in a certain city where the basic rate is $4 per month. The officials have evidence that if the charge is reduced the number of phones will increase at an estimated rate of 1,000 phones for each 25 cent reduction. What rate would yield the greatest income?

2) What would be the most favorable rate in the problem above if it must be assumed that each added phone increases operating costs by 50 cents per month?

Answer 1

For #1,

Income = (rate)(# of phones)
I = (4-.25x)(10000+1000x)
I = -250x²+1500x+40000
(take derivative, and set = 0)
I' = -500x+1500 = 0
500x=1500
x=3
rate = 4-.25(3) = $3.25/month