To find the system of inequalities that describes the constraints on the number of apple tarts [tex]\( t \)[/tex] and apple pies [tex]\( p \)[/tex] that a baker can make each day given the scenario described, let's break it down step-by-step.
1. Limit on Number of Tarts:
- The baker can make no more than 40 tarts per day.
[tex]\[ t \leq 40 \][/tex]
2. Constraint on Apple Usage for Tarts:
- Each tart requires 1 apple.
- Therefore, the inequality representing the maximum apple usage:
[tex]\[ t \leq 184 \][/tex]
3. Reiteration of Limit on Number of Tarts:
- Since the constraint on tarts is an independent condition:
[tex]\[ t \leq 40 \][/tex]
4. Constraint on Apple Usage for Pies Alone:
- Each pie requires 8 apples.
- Therefore, the inequality representing the maximum apple usage for pies alone:
[tex]\[ 8p \leq 184 \][/tex]
5. Reiteration of Limit on Number of Tarts:
- This is an independent condition given, so:
[tex]\[ t \leq 40 \][/tex]
6. Combined Apple Usage for Both Tarts and Pies:
- The total number of apples used by both tarts and pies combined should not exceed 184 apples.
- Therefore:
[tex]\[ p + 8t \leq 184 \][/tex]
7. Reiteration of Limit on Number of Tarts:
- This is an independent condition given:
[tex]\[ t \leq 40 \][/tex]
8. Combined Constraint on Both Pies and Tarts:
- You need to consider both tarts and pies together.
- The total limit of the number of apples used by both pies and tarts should not exceed 184 apples, considering the number of pies made:
[tex]\[ 8p + t \leq 184 \][/tex]
Thus, the system of inequalities that can be used to find the possible number of pies and tarts the baker can make is:
[tex]\[
\begin{aligned}
t & \leq 40 \\
p & \leq 184 \\
t & \leq 40 \\
8 p & \leq 184 \\
t & \leq 40 \\
p + 8 t & \leq 184 \\
t & \leq 40 \\
8 p + t & \leq 184
\end{aligned}
\][/tex]
