An amusement park charges $8 admission and average of 2000 visitors per day. A survey shows that for each $1 increase in the admission cost, 100 fewer people would visit the park.
a) Write an equation to express the revenue, R(x) dollars, in terms of a price increase of x dollars
b) Find the coordinates of the maximum point of this function
c) What admission cost gives the maximum revenue?
d) How many visitors give the maximum revenue
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You need to find the revenue, R(x), which is by definition,
R(x) = (admission per person) x (number of people)
We know that if we increase the admission per person by 1 unit, the number of people goes down 1 unit, so
R(x) = (8 + 1)*(2000 - 100)
If we define the variable 'x' to represent the number of increases in price, from what we've begun to establish,
\[R(x)=(8+x)(2000-100x)\]Expanding gives, \[R(x)=16000+1200x-200x^2\]
The maximum point will be given for\[R'(x)=1200-200x\]Setting R'(x) to zero and solving gives,\[x=6\]The corresponding R-coordinate is then, \[R(6)=19600\]The coordinate of the maximum is therefore (6,19600).
The park should make a 6-unit increase to maximize its revenue. So the admission cost should be $14.
The number of people attending would then by 19,600.