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

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

Sagot :

GinnyAnswer
To solve this problem, let's break down the given information step by step.

1. Defining Variables
- Let [tex]\( t \)[/tex] represent the number of apple tarts made each day.
- Let [tex]\( p \)[/tex] represent the number of apple pies made each day.

2. Apple Requirements
- Each apple tart requires 1 apple.
- Each apple pie requires several apples. Let's denote this amount by [tex]\( a \)[/tex].

3. Supplies and Constraints
- The baker receives a shipment of 184 apples every day.
- The baker can make no more than 40 tarts each day.
- We need to set up a system of inequalities to reflect these constraints and limitations.

From the problem, here's what we know:

1. Constraint on tarts:
- The number of tarts, [tex]\( t \)[/tex], is limited to 40:
[tex]\[ t \leq 40 \][/tex]

2. Constraint on the total apples used:
- Since each tart requires 1 apple and each pie requires [tex]\( a \)[/tex] apples, the total number of apples used by tarts and pies must not exceed 184 apples. This gives us the inequality:
[tex]\[ 1t + ap \leq 184 \][/tex]

Now, let's clarify a potential way to express these constraints:

- Given that [tex]\( a \)[/tex] is a certain number of apples each pie use (the typical value given in several context is 8 apples per pie, therefore we can assume [tex]\( a = 8 \)[/tex]).
- The total number of apples used by the pies would be [tex]\( 8p \)[/tex], and the total number of apples used by the tarts is [tex]\( 1t \)[/tex].

Putting it all together, we get:
[tex]\[ t \leq 40 \][/tex]
[tex]\[ p \leq 184 \][/tex] (due to a constraint on total apple pie production despite \(1 \leq 184 typically unnecessary in practice but for ensuring constraints non-violation, p limitless)
[tex]\[ 8p \leq 184 \][/tex]
[tex]\[ 1t + 8p \leq 184 \][/tex]
Hence the feasible combination of constraints that can be adopted from are based on necessary constraints planning agent, hence (as pies more typically pie requirement of \(t\geq 0 and non-excesive upon manual work constraint being in context reduces apple-wise):

Thus the final sets of inequalities for valid combination;
- \(\boxed{
\begin{aligned}
t & \leq 40 \\
8p & \leq 184 \\
t + 8p & \leq 184
\end{aligned}
}$
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