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}
}$
