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.
Write an equation to express the revenue, R(x) dollars, in terms of a price increase of x dollars
Find the coordinates of the maximum point of this function
What admission cost gives the maximum revenue?
How many visitors give the maximum revenue?
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y=4x+8
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Sure, I can help you with that using my humorous approach! Let's get started.
To find an equation for the revenue, R(x), in terms of a price increase of x dollars, we need to consider the given information.
First, we know that the admission price is initially $8, and the average number of visitors per day is 2000. For each $1 increase in the admission cost, 100 fewer people would visit the park.
So, if we let "p" represent the admission price, then the number of visitors can be expressed as (2000 - 100x), where "x" is the increase in dollars. Therefore, the revenue can be calculated by multiplying the number of visitors with the admission price as follows:
R(x) = p * (2000 - 100x)
To find the coordinates of the maximum point of this function, we need to determine the value of "x" that maximizes the revenue. One way to do this is by finding the vertex of the parabolic revenue equation.
The formula for the x-coordinate of the vertex of a quadratic function in vertex form is given as: x = -b/2a
In our case, the equation for R(x) is not in the standard vertex form (ax^2 + bx + c), but we can rewrite it by rearranging the terms and putting it in the form: R(x) = -100x^2 + 2000p
Now, let's identify a and b coefficients in the equation: a = -100 and b = 0.
Using the formula x = -b/2a, we can substitute the values and find the x-coordinate of the vertex:
x = -0 / (2 * -100) = 0
Therefore, the x-coordinate of the vertex, or the maximum point of the revenue function, is x = 0.
To find the admission cost that gives the maximum revenue, we can substitute this value of x = 0 back into the equation for R(x):
R(0) = p * (2000 - 100 * 0)
R(0) = p * 2000
This shows that when there is no increase in the admission price (x = 0), the revenue is simply the admission price multiplied by the average number of visitors per day, which is 2000.
Finally, to determine the number of visitors that give the maximum revenue, we can substitute x = 0 into the equation for the number of visitors:
Visitors = 2000 - 100 * 0
Visitors = 2000
So, at the admission cost that gives the maximum revenue, there will be 2000 visitors per day.
I hope this helps! If you have any more questions or need further assistance, feel free to ask.
To find an equation to express the revenue, R(x) dollars, in terms of a price increase of x dollars, we need to consider the given information.
We know that the admission cost is initially $8, and for each $1 increase in the admission cost, 100 fewer people would visit the park. This means that if we increase the admission price by x dollars, the number of visitors would decrease by 100x.
Initially, the park has 2000 visitors per day. So, if we increase the admission price by x dollars, the number of visitors would be 2000 - 100x.
To calculate the revenue, we multiply the admission price by the number of visitors. The admission price is $8 + x, and the number of visitors is 2000 - 100x. Therefore, the revenue can be calculated as:
R(x) = (8 + x)(2000 - 100x)
To find the coordinates of the maximum point of this function, we need to determine the value of x that will maximize the revenue. We can do this by taking the derivative of the revenue function and setting it equal to zero, then solving for x.
Let's differentiate R(x) with respect to x:
R'(x) = [(8 + x)(2000 - 100x)]'
To simplify, we can use the product rule:
R'(x) = [2000(8 + x) - (8 + x)(100x)]'
Simplifying further:
R'(x) = 2000 - 100x - 100x
Now, setting R'(x) equal to zero and solving for x:
2000 - 100x - 100x = 0
-200x + 2000 = 0
-200x = -2000
x = -2000 / -200
x = 10
So, the value of x that maximizes the revenue is 10 dollars.
To find the admission cost that gives the maximum revenue, we need to add the value of x to the initial admission cost of $8:
Admission cost = $8 + x
Admission cost = $8 + 10
Admission cost = $18
Therefore, an admission cost of $18 would give the maximum revenue.
To determine the number of visitors that give the maximum revenue, we substitute the value of x = 10 into the number of visitors equation:
Number of visitors = 2000 - 100x
Number of visitors = 2000 - 100(10)
Number of visitors = 2000 - 1000
Number of visitors = 1000
Therefore, 1000 visitors would give the maximum revenue.
R(x) = (8+x)(2000-100x)
So, that is just a parabola, with its vertex (maximum revenue) at x=6
Now you can answer the other questions.