Let's break down the given scenario step by step to understand the values represented by the polynomial expression [tex]\( P(x) = -40x^2 - 100x + 27,500 \)[/tex].
1. Understanding the Constant Term (27,500):
- The amusement park prices tickets at \[tex]$55 each.
- On average, 500 tickets are sold daily.
- The initial income without any ticket price changes can be calculated as:
\[
55 \text{ (price per ticket)} \times 500 \text{ (average tickets sold)} = 27,500
\]
- Thus, the constant term 27,500 represents the initial income before any changes in ticket prices and quantities sold are accounted for.
2. Understanding the Binomial Term \((500 - 20x)\):
- Each $[/tex]2 increase in ticket price ([tex]\(x\)[/tex] increments) results in 20 fewer tickets sold.
- Let [tex]\(x\)[/tex] be the number of [tex]$2 increases.
- The number of tickets sold after \(x\) increments is:
\[
500 \text{ (initial average tickets)} - 20x \text{ (reduction based on increments)}
\]
- The binomial expression \((500 - 20x)\) represents the change in the number of tickets sold based on the number of $[/tex]2 increases.
Therefore, the constant of the polynomial expression 27,500 represents the initial income before any changes in the price of a ticket. The binomial [tex]\((500 - 20x)\)[/tex] represents the change in the number of tickets sold based on [tex]\(x\)[/tex] increases in the price of a ticket.
