You operate a gaming website, where users must pay a small fee to log on. when you charged $4 the demand was 540 log-ons per month. when you lowered the price to $3.50, the demand increased to 810 log-ons per month. (a) construct a linear demand function for your website and hence obtain the monthly revenue r as a function of the log-on fee x. r(x) = (b) your internet provider charges you a monthly fee of $30 to maintain your site. express your monthly profit p as a function of the log-on fee x. hint [see example 4.] p(x) = determine the log-on fee you should charge to obtain the largest possible monthly profit. x = $ what is the largest possible monthly profit? $

A) Demand function

price (x)          demand (D(x))

4                      540

3.50                  810

D - 540          810 - 540
= 
x - 4               3.50 - 4

D - 540
= - 540
x - 4

D - 540 = - 540(x - 4)

D = -540x + 2160 + 540

D = 2700 - 540x

D(x) = 2700 - 540x

Revenue function, R(x)

R(x) = price * demand = x * D(x)

R(x) = x* (2700 - 540x) = 2700x - 540x^2

b) Profit, P(x)

profit = revenue - cost

P(x) = R(x) - 30

P(x) = [2700x - 540x^2] - 30

P(x) = 2700x - 540x^2 - 30

Largest possible profit => vertex of the parabola

vertex of 2700x - 540x^2 - 30

When you calculate the vertex you find x = 5 /2

=> P(x) = 3345

you should charge a log-on fee of $2.5 to have the largest profit, which is $3345.