An amusement park charges $8 admission and averages 2000 visitors per day. A survey shows that, for each $1 increase in admission cost, 100 fewer people would visit the park. Find the admission cost and number of visitors that gives the maximum revenue.

Question

An amusement park charges $8 admission and averages 2000 visitors per day. A survey shows that, for each $1 increase in admission cost, 100 fewer people would visit the park. Find the admission cost and number of visitors that gives the maximum revenue.

Answer 1

see your previous post. It works just the same way.

Answer 2

To find the admission cost and number of visitors that give the maximum revenue, we need to consider the relationship between the admission cost and the number of visitors.

Let's assume x represents the number of $1 increases in the admission cost. For example, if x = 0, the admission cost is $8. If x = 1, the admission cost would be increased by $1, resulting in $9, and so on.

Given that for each $1 increase in admission cost, 100 fewer people would visit the park, we can express the number of visitors as 2000 - 100x.

To calculate the revenue, we multiply the admission cost by the number of visitors: Revenue = Admission Cost * Number of Visitors.

Therefore, the revenue can be expressed as: Revenue = (8 + x) * (2000 - 100x).

To find the maximum revenue, we need to find the value of x that maximizes this function. We can do this by finding the vertex of the quadratic equation representing the revenue function.

The vertex of the quadratic equation can be found using the formula: x = -b / (2a), where a is the coefficient of the x^2 term, b is the coefficient of the x term, and c is the constant term.

In this case, the equation representing the revenue function is: Revenue = -100x^2 + 1800x + 16000.

Comparing this to the standard quadratic form ax^2 + bx + c, we can see that a = -100, b = 1800, and c = 16000.

Now we can find the value of x that gives the maximum revenue:

x = (-b) / (2a) = (-1800) / (2 * (-100)) = -1800 / (-200) = 9.

This means that when x = 9, we have the maximum revenue.

Substituting x = 9 into the equation for the number of visitors, we can find the number of visitors:

Number of Visitors = 2000 - 100x = 2000 - 100 * 9 = 2000 - 900 = 1100.

So, to maximize revenue, the admission cost would be $8 + $9 = $17, and the number of visitors would be 1100.