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Studies have shown that 500 people will attend a high school basketball game when the
admission price is $2. In the championship game admission price will increase. For every
$2 increase, 20 fewer people will attend.
a) What is the number of $2 increases in price that will maximize the revenue?
b) What price per ticket will maximize the revenue?
c) What is the maximum revenue?

Respuesta :

The number of increases is 12, the price per ticket that maximize the revenue is $26 and the maximum revenue is $169.

Equation formation

To solve the question, it is necessary to form an equation that represents the sentences spoken.

First, we will set up the revenue equation, which will be equal to the number of people multiplied by the ticket price, so that:

                                           [tex]R = N \times P[/tex]

We know that with each increase made, 20 people decrease, so we can represent the number of people and the ticket price in such a way:

[tex]N = 500 - 20x\\P = 2 + 2x[/tex]

Thus, we have the following expression:

                                     [tex]R = (500-20x)(2+2x)[/tex]

                                     [tex]R = -40^{2}+960x+1000[/tex]

                                       [tex]R = -x^{2} + 24x+25[/tex]

Now, to find the number of increases performed that would maximize the value of revenue, we need to find the value of the X of the vertex (value of X where Y is maximum), having:

                                                [tex]X_v = \frac{-b}{2a}[/tex]

                                              [tex]X_v = \frac{-24}{-2} = > 12[/tex]

So,  the number of increases is equal to 12 ans the price per ticket will be equal to:

                                          [tex]P = 2 + 2x\\P = 2 + 2\times 12\\P = 26[/tex]

Finally, to calculate the maximum revenue, simply substitute the value of x found in the equation, having:

                                   [tex]R = -144 + 288 + 25 = > 169[/tex]

So, the answer will be:

a) 12 increases

b) $26

c) $169

Learn more about X and Y of the vertex: brainly.com/question/4332303                    

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