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.
