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An amusement park prices tickets at \[tex]$55 and sells an average of 500 tickets daily. The management finds, over multiple increases in ticket pricing, 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[/tex] to model the daily earnings of the amusement park, where [tex]x[/tex] is the number of \$2 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 daily earnings when there are no price increases. [tex]\[\boxed{27,500}\][/tex]

The binomial [tex]\((500 - 20x)\)[/tex] is a factor of the polynomial expression and represents the decrease in the number of tickets sold due to price increases.

Sagot :

GinnyAnsweridn
Let's break down the given polynomial expression [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex] and analyze its components to answer the question:

1. Constant Term in the Polynomial Expression:
- The polynomial given is [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex].
- The constant term here is [tex]\( 27,500 \)[/tex].
- Since this constant term represents a value independent of [tex]\( x \)[/tex], in the context of the problem, it denotes the daily earnings without any price increases.

2. The Binomial [tex]\((500 - 20x)\)[/tex]:
- This represents the number of tickets sold.
- Initially, 500 tickets are sold at the base price.
- For each [tex]\( \$2 \)[/tex] increase in price (represented by [tex]\( x \)[/tex]), 20 fewer tickets are sold.
- So, the term [tex]\((500 - 20x)\)[/tex] directly adjusts the number of tickets sold according to the price increase.

Using this understanding, we can complete the sentences:

1. The constant of the polynomial expression represents the daily earnings with 0 price increases.
2. The binomial [tex]\((500 - 20x)\)[/tex] is a factor of the polynomial expression and represents the number of tickets sold.

Thus, the completed sentences are:

- The constant of the polynomial expression represents the daily earnings with 0 price increases.
- The binomial [tex]\((500-20x)\)[/tex] is a factor of the polynomial expression and represents the number of tickets sold.
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