Let’s break the problem down using the given information and the provided answer.
1. Understanding the Polynomial Expression:
The polynomial given is:
[tex]\[
P(x) = -40x^3 - 100x + 27,500
\][/tex]
The constant term in this polynomial represents a key value. The constant term of the polynomial is 27,500.
2. Identifying the Binomial Factor:
The binomial given is [tex]\(600 - 20r\)[/tex]. This binomial is treated as a factor of the polynomial related to ticket pricing and revenue generation.
3. Initial Daily Earnings Calculation:
The initial price per ticket is \[tex]$55, and the average number of tickets sold daily is 60. The initial daily earnings are therefore calculated by multiplying these two values:
\[
55 \, \text{(dollars per ticket)} \times 60 \, \text{(tickets per day)} = 3300 \, \text{(dollars per day)}
\]
4. Daily Earnings After One Increment:
After one increment (or increase) in ticket price, the new price per ticket is given to be after 12 increments. Initially, each increment appears to be \$[/tex]1 more (as increments are typically small increases).
The new price calculation after 12 increments:
[tex]\[
55 + 12 = 67 \, \text{(dollars per ticket)}
\][/tex]
With the same number of tickets sold (60 tickets), the new daily earnings are:
[tex]\[
67 \, \text{(dollars per ticket)} \times 60 \, \text{(tickets per day)} = 4020 \, \text{(dollars per day)}
\][/tex]
Given these observations, now we fill in the sentences based on the provided context:
- The constant of the polynomial expression represents the value 27,500.
- The binomial [tex]\(600-20r\)[/tex] is a factor related to the polynomial and price of a ticket.
- Initial daily earnings of the amusement park before any increases are [tex]$3,300.
- Daily earnings of the amusement park after one 12 increment increase is $[/tex]4,020.
Note: The question regarding the number of tickets sold after four increments can’t be conclusively determined without further specific information, so it is left unanswered here.
