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
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
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.