Let's analyze the given situation and find the appropriate answers for each point in the context of the system of inequalities provided.
We have two inequalities that define the system:
1. [tex]\( r \leq -0.23 p^2 + 2.25 p \)[/tex]
2. [tex]\( r \geq 5 \)[/tex]
These inequalities represent the revenue [tex]\( r \)[/tex] in tens of thousands of dollars (i.e., [tex]\( r = 5 \)[/tex] represents [tex]\(\$ 50,000\)[/tex]) and the price of popcorn [tex]\( p \)[/tex] in dollars. The goal is to identify whether the given points satisfy both inequalities.
### Point (4, 6)
For the point [tex]\((4, 6)\)[/tex]:
- [tex]\( p = 4 \)[/tex]
- Calculating [tex]\( r \)[/tex]:
[tex]\[
r = -0.23 (4)^2 + 2.25 (4) = -0.23 (16) + 9 = -3.68 + 9 = 5.32
\][/tex]
- Check if [tex]\( r \leq 6 \)[/tex]:
[tex]\[
5.32 \leq 6 \quad (\text{True})
\][/tex]
- Check if [tex]\( r \geq 5 \)[/tex]:
[tex]\[
5.32 \geq 5 \quad (\text{True})
\][/tex]
Thus, the point [tex]\( (4, 6) \)[/tex] satisfies both inequalities.
### Point (6, 5)
For the point [tex]\((6, 5)\)[/tex]:
- [tex]\( p = 6 \)[/tex]
- Calculating [tex]\( r \)[/tex]:
[tex]\[
r = -0.23 (6)^2 + 2.25 (6) = -0.23 (36) + 13.5 = -8.28 + 13.5 = 5.22
\][/tex]
- Check if [tex]\( r \leq 5 \)[/tex]:
[tex]\[
5.22 \leq 5 \quad (\text{False})
\][/tex]
- Check if [tex]\( r \geq 5 \)[/tex]:
[tex]\[
5.22 \geq 5 \quad (\text{True})
\][/tex]
However, since the first inequality is not satisfied, the point [tex]\( (6, 5) \)[/tex] does not meet the conditions of the system.
In summary:
- The point [tex]\((4, 6)\)[/tex] is a solution of this system.
- The point [tex]\((6, 5)\)[/tex] is not a solution of this system.
