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An amusement park prices tickets at [tex]$\$[/tex] 55[tex]$ and sells an average of 500 tickets daily. The management finds, over multiple increases in ticket pricing, that a $[/tex]\[tex]$ 2$[/tex] increase in the price of a ticket leads to an average of 20 fewer tickets being sold in a day.

Management uses the combined function [tex]$P$[/tex] to model the daily earnings of the amusement park, where [tex]$x$[/tex] is the number of [tex]$\$[/tex] 2[tex]$ increases in the price of a ticket.
$[/tex]
P(x) = -40x^2 - 100x + 27,500
[tex]$

Use the given information to complete the sentences.
The constant of the polynomial expression represents the $[/tex]\square[tex]$ in the price of a ticket.
The binomial $[/tex](500 - 20x)[tex]$ is a factor of the polynomial expression and represents the $[/tex]\square$ in the price of a ticket.

Sagot :

GinnyAnsweridn
To solve this problem, we will analyze the components of the polynomial expression modeling the daily earnings of the amusement park, given by:

[tex]\[ P(x) = -40x^2 - 100x + 27,500 \][/tex]

The questions ask us to identify what the constant term and the binomial [tex]\((500 - 20x)\)[/tex] represent in the context of ticket pricing.

1. The constant of the polynomial expression:

The constant term in a polynomial expression is the value that does not change when the variable (in this case, [tex]\(x\)[/tex]) changes. Here, the constant term is 27,500. This value represents the daily earnings of the amusement park when there are no \[tex]$2 increases in ticket prices (i.e., when \(x = 0\)). Therefore, the constant \(27,500\) represents the original daily earnings without any increases in the price of a ticket. 2. The binomial \((500 - 20x)\): The binomial expression \((500 - 20x)\) within the polynomial represents the adjustment in the number of tickets sold per day based on the number of \$[/tex]2 increases.

Specifically, when [tex]\(x\)[/tex] represents the number of \[tex]$2 increases, each increase will result in selling 20 fewer tickets. The initial number of tickets sold daily is 500. The term \(20x\) indicates a reduction of 20 tickets for each \$[/tex]2 increase (i.e., for each increase in [tex]\(x\)[/tex]).

Therefore, the binomial [tex]\((500 - 20x)\)[/tex] represents the number of tickets sold in a day after [tex]\(x\)[/tex] \$2 increases in the price of a ticket.

Using this detailed analysis, we can accurately complete the sentences as follows:

- The constant of the polynomial expression represents the original daily earnings in the price of a ticket.
- The binomial [tex]\((500 - 20x)\)[/tex] is a factor of the polynomial expression and represents the number of tickets sold in a day.
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