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An amusement park prices tickets at \[tex]$55 and sells an average of 500 tickets daily. The management finds that a \$[/tex]2 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]\( P(x) \)[/tex] to model the daily earnings of the amusement park, where [tex]\( x \)[/tex] is the number of \[tex]$2 increases in the price of a ticket.
\[ P(x) = -40x^2 - 100x + 27,500 \]

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 \((500 - 20x)\) is a factor of the polynomial expression and represents the $[/tex]\square$ in the price of a ticket.

Sagot :

GinnyAnswer
Let's break down the question step by step:

1. We start with the polynomial expression [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex] which models the daily earnings of the amusement park. In this expression:
- [tex]\( x \)[/tex] represents the number of [tex]\( \$2 \)[/tex] increases in the price of a ticket.

2. Understanding the terms in the polynomial:
- The constant term is [tex]\( 27,500 \)[/tex]. This term represents the initial daily earnings before any price increase.
- The binomial expression [tex]\( (500-20x) \)[/tex] is mentioned as a factor of the polynomial. Let's interpret this:
- [tex]\( 500 \)[/tex] stands for the average number of tickets sold when no price increase is applied.
- [tex]\( 20x \)[/tex] accounts for the decrease in the number of tickets sold when the price of a ticket increases.

Given these observations, we can fill out the sentences as follows:
1. The constant of the polynomial expression represents the initial daily earnings in the price of a ticket.
2. The binomial [tex]\((500 - 20x)\)[/tex] is a factor of the polynomial expression and represents the number of tickets sold based on the new price in the price of a ticket.

Therefore, the correct answers should be:

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