You operate a gaming Web site, www.mudbeast.net, where users must pay a small fee to log...

 
  

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You operate a gaming Web site, www.mudbeast.net, where users must pay a small fee to log on. When you charged $4 the demand was 570 log-ons per month. When you lowered the price to $3.50, the demand increased to 855 log-ons per month. (a) Construct a linear demand function for your Web site and hence obtain the monthly revenue R as a function of the log-on fee x. Solution: To construct the linear demand function q=mx+b, use the given information to find the slope m 92-9855-570 3.5-4 285 -0.5 <=-570. Then, use the slope to find beq=-570x+b, and 570-570(4) + b. so b = 2850. The demand function is q=-570x + 2850. Next, recall that Revenue = (price) x ( x (quantity), so R=xq=x(-570x+2850)=-570x + 2850x. (b) 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 Revenue-Cast, and the only cost associated with this problem Solution: Recall that Profit the $40 monthly fee. Thus, Determine the log-on fee you should charge to obtain the largest possible monthly profit. Solution: The profit function is quadratic, with leading coefficient that's negative. Therefore, it opens downwand and its vertex is the maximum paint. So, we must find the x-coordinate of the vertex to find the log-on fee that will maximize the monthly profit. Use the vertex formula to do this (found on p. 621 of the textbook): -2850 Fan2-570) =25 Thus, they should charge $2.50 What is the largest possible monthly profit? Solution: Here we need to find the P-coordinate of the vertex. Do this by plugging the x- coordinate back into the profit function P=-570(25) +2850(25)-40-35225 Thus, the largest monthly profit is $3.522.50 P=-570x² +2850x-40 You operate a gaming Web site, www.mudbeast.net, where users must pay a small fee to log on. When you charged $4 the demand was 570 log-ons per month. When you lowered the price to $3.50, the demand increased to 855 log-ons per month. (a) Construct a linear demand function for your Web site and hence obtain the monthly revenue R as a function of the log-on fee x. Solution: To construct the linear demand function q=mx+b, use the given information to find the slope m 92-9855-570 3.5-4 285 -0.5 <=-570. Then, use the slope to find beq=-570x+b, and 570-570(4) + b. so b = 2850. The demand function is q=-570x + 2850. Next, recall that Revenue = (price) x ( x (quantity), so R=xq=x(-570x+2850)=-570x + 2850x. (b) 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 Revenue-Cast, and the only cost associated with this problem Solution: Recall that Profit the $40 monthly fee. Thus, Determine the log-on fee you should charge to obtain the largest possible monthly profit. Solution: The profit function is quadratic, with leading coefficient that's negative. Therefore, it opens downwand and its vertex is the maximum paint. So, we must find the x-coordinate of the vertex to find the log-on fee that will maximize the monthly profit. Use the vertex formula to do this (found on p. 621 of the textbook): -2850 Fan2-570) =25 Thus, they should charge $2.50 What is the largest possible monthly profit? Solution: Here we need to find the P-coordinate of the vertex. Do this by plugging the x- coordinate back into the profit function P=-570(25) +2850(25)-40-35225 Thus, the largest monthly profit is $3.522.50 P=-570x² +2850x-40

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