Encouraged by the popularity of your Dungeons and Dragons website, www.mudbeast.net, you have decided to charge users who log on to the site. When you charged a $1.50 access fee, your web counter showed a demand of 270 "hits" per month. After you lowered the price to $0.50, activity increased to 350 "hits" per month. Obtain the monthly revenue R as a function of the access fee x.
A) $R = - 80 x ^ { 2 } - 390 x$
B) $R = 80 x ^ { 2 } - 230 x$
C) $R = - 80 x ^ { 2 } + 390 x$
D) $R = - 80 x + 390$
E) $R = 80 x - 230$

Question

Encouraged by the popularity of your Dungeons and Dragons website, www.mudbeast.net, you have decided to charge users who log on to the site. When you charged a $1.50 access fee, your web counter showed a demand of 270 "hits" per month. After you lowered the price to $0.50, activity increased to 350 "hits" per month. Obtain the monthly revenue R as a function of the access fee x. 
A) $R = - 80 x ^ { 2 } - 390 x$
B)  $R = 80 x ^ { 2 } - 230 x$
C)  $R = - 80 x ^ { 2 } + 390 x$
D)  $R = - 80 x + 390$
E)  $R = 80 x - 230$

Quizplus · Accepted Answer

To find the monthly revenue $R$ as a function of the access fee $x$, we can use the given points to form a linear equation. The points given are (1.5, 270) and (0.5, 350), where the first number in each pair is the access fee $x$ and the second number is the number of hits, which when multiplied by the access fee gives the revenue $R$. First, we calculate the slope ($m$) of the line connecting these two points:$$m = \frac{350 - 270}{0.5 - 1.5} = \frac{80}{-1} = -80$$This tells us that for every dollar increase in the access fee, the revenue changes by -80 units. However, to find the exact formula for $R$, we need to use one of the points to solve for the y-intercept ($b$) in the equation $R = mx + b$. Using the point (1.5, 270):$$270 = -80(1.5) + b$$$$270 = -120 + b$$$$b = 390$$Therefore, the monthly revenue $R$ as a function of the access fee $x$ is:$$R = -80x + 390$$However, the correct formulation of the revenue function should account for the quadratic nature implied by the change in demand with price, which is not directly addressed in the linear approach above. The correct answer, based on the options provided and considering a mistake in the linear versus quadratic distinction, would be closest to the option that reflects a change in revenue with price, but none of the options perfectly match a linear model derived from the given points. The closest match given the context and the need for a correction in explanation is $R = - 80 x ^ { 2 } + 390 x$, which mistakenly was approached linearly but should reflect a quadratic relationship due to the nature of price elasticity and demand. This option, however, is not directly derivable from the simplistic linear approach initially described and reflects a misunderstanding in the explanation. The correct approach to derive such a quadratic equation would involve more complex considerations of price elasticity and demand not directly addressed with the simple slope calculation. Thus, the selection of C is based on the options provided and the context, but with an acknowledgment of the error in the explanation process.

Encouraged by the Popularity of Your Dungeons and Dragons Website R=−80x2−390xR = - 80 x ^ { 2 } - 390 xR=−80x2−390x

Encouraged by the Popularity of Your Dungeons and Dragons Website $R = - 80 x ^ { 2 } - 390 x$