You operate a gaming wabsite, www.mudbeast.net, where users must pay a smal fee

Question

You operate a gaming wabsite, www.mudbeast.net, where users must pay a smal fee to log an. When you charged $1 the demand was ss0 og-ons per month. When you lowered the price to 3.50, the demend increased to 825 log-ons per month. (e) Construct a linear demand function for your website and hence obtain the monthly revenue R as a function of the log-on feex R(x) (tb) Your Internet provider charges you a monthly fee of $40 to maintain your site. Express your monthly profit P as a function of the log-on fee x. HINT [See Example 4.] Determine the log-on fee you should charge to obtain the largest possible monthly profit x=$35 What is the largest possible monthly profit?

Explanation / Answer

a)

To construct the linear demand function D = mx + c, use the given information

to find the slope m = (D2 - D1)/(x2 - x1)
=(825 550)/(3.5 4)
=275/-0.5
= -550
Then, use the slope to find c:

D = 550x + c

550 = 550(4) + c

c= 2750. The demand function is D = 550x + 2750

Revenue = (price) × (quantity)

R= xD = x(550x + 2750)

R= 550x^2 + 2750x -------(1)

b)

Profit = Revenue Cost

P = -550x^2 +2750x - 40 ---------(2)

c)

Differentiate equation(2) with respect to x

dp/dx = -550(2x) + 2750*1 - 0

=-1100x + 2750

for max or minima

dp/dx = 0

-1100 x +2750 = 0

x = (2750)/(1100) = 2.5

largest monthly profit

Pmax = -550(2.5)^2 + 2750(2.5) -40

=3397.5

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