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A baker makes apple tarts and apple pies each day. Each tart, [tex]\( t \)[/tex], requires 1 apple, and each pie, [tex]\( p \)[/tex], requires 8 apples. The baker receives a shipment of 184 apples every day. If the baker makes no more than 40 tarts per day, which system of inequalities can be used to find the possible number of pies and tarts the baker can make?

[tex]\[
\begin{aligned}
t & \leq 40 \\
p & \leq 184 \\
8p & \leq 184 \\
p + 8t & \leq 184 \\
8p + t & \leq 184
\end{aligned}
\][/tex]



Answer :

GinnyAnswer
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]

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