The dance company has a $10 cover charge, an average of 200 people will attend. For each $0.25 increase in admission price, the average number attending decreases by 1. What should the owner charge in order to make the most money? what is the maximum money that can be earned?

Question

The dance company has a $10 cover charge, an average of 200 people will attend. For each $0.25 increase in admission price, the average number attending decreases by 1. What should the owner charge in order to make the most money? what is the maximum money that can be earned?

Answer 1

If the price is 10.00 + .25x

attendance = 200 - x
money = attendance * price = (200-x)(10.00+.25x)
= -.25x^2 + 40x + 2000

this is a parabola with vertex at x = 40/.5 = 80

at x=80, admission is $30, money = 3600

Answer 2

To determine the admission price that would generate the most revenue for the dance company, we need to find the point where the product of the admission price and the number of attendees is maximized.

Let's analyze the given information step by step:

1. The base admission price is $10, and the average number of attendees is 200.
2. For every $0.25 increase in admission price, the average number of attendees decreases by 1.

To solve this problem, we can create a revenue function that calculates the total revenue based on the admission price and the number of attendees. Let's denote the admission price as "x" (in dollars) and the number of attendees as "y." The revenue function can be represented as follows:

Revenue = x * y

Now, let's break down the problem further. We know that the average number of attendees decreases by 1 for every $0.25 increase in price, so we can express the number of attendees as a function of the admission price:

y = 200 - 4(x - 10)

Simplifying this equation, we get:

y = 200 - 4x + 40
y = -4x + 240

Substituting this value for "y" in the revenue function, we get:

Revenue = x * (-4x + 240)
Revenue = -4x^2 + 240x

To find the admission price that maximizes revenue, we need to find the vertex of the quadratic function. The vertex of a quadratic function can be calculated using the formula:

x = -b / (2a)

In our case, a = -4 and b = 240. Substituting these values into the formula, we get:

x = -240 / (2 * -4)
x = -240 / -8
x = 30

Therefore, to maximize revenue, the owner should set the admission price at $30. The maximum revenue can be calculated by substituting the value of "x" back into the revenue function:

Revenue = -(4 * 30^2) + (240 * 30)
Revenue = -3600 + 7200
Revenue = $3600

Hence, the dance company should charge $30 for admission in order to generate the most revenue, and the maximum revenue that can be earned is $3600.