The latest demand equation for your gaming Web site wwwmudbe

The latest demand equation for your gaming Web site, www.mudbeast.net, is given by

q = 500x + 1000

where q is the number of users who log on per month and x is the log-on fee you charge. Your Internet provider bills you as follows:

Find the monthly cost as a function of the log-on fee x.

Find the monthly profit as a function of x.

Determine the log-on fee you should charge to obtain the largest possible monthly profit.
x = $  per log-on

What is the largest possible monthly profit?
$

Solution

q = -500x +1000

q ---> number of users
x ---> log on fee

Access fee is 0.40 dollars per log on
number of users = q

So, total access fee is : 0.4q dollars
= 0.4(-500x + 1000)
= -200x + 400

Total fee is : 10 + (-200x + 400)
C = -200x + 410 ---> ANSWER

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Revenue is x(-500x + 1000)
R = -500x^2 + 1000x

Profit = R - C

P = (-500x^2 + 1000x) - (-200x + 410)

P = -500x^2 + 1200x - 410 ----> ANSWER

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MAx profit :
dP/dx = 0
-1000x + 1200 = 0
1000x = 1200
x = 1200/1000
x = 1.2 dollars is the log on fee -----> ANSWER

Max profit becomes :
Pmax = -500x^2 + 1200x - 410 where x = 1.2
Pmax = -500(1.2)^2 + 1200(1.2) - 410
Pmax = -720 + 1440 - 410
Pmax = 310 dollars -----> ANSWER