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Sagot :
Let's break down the problem step by step to determine the correct inequalities and then identify which combinations of small prints and large prints satisfy these constraints.
### Step 1: Define the Inequalities
We have two conditions to consider:
1. Profit Condition:
The store needs to make at least $700 per day. The profit is calculated based on the small prints (\(x\)) and large prints (\(y\)):
[tex]\[ 15x + 25y \geq 700 \][/tex]
2. Production Condition:
The number of small prints (\(x\)) must be more than three times the number of large prints (\(y\)):
[tex]\[ x > 3y \][/tex]
Given these inequalities, we can identify the correct options from the provided table.
- \(15x + 25y \geq 700\)
- \(x > 3y\)
Now let's match these with the given options:
[tex]\[ \begin{array}{|c|c|c|} \hline \multicolumn{3}{|c|}{Inequality Options} \\ \hline x+y \leq 60 & 15x + 25y < 700 & x > 3y \\ \hline 15x + 25y \geq 700 & y > 3x & x + 3y \geq 60 \\ \hline \end{array} \][/tex]
From this table, the correct inequalities are:
- \(15x + 25y \geq 700\)
- \(x > 3y\)
### Step 2: Verify the Combinations
Now, we need to check which combinations of small prints (\(x\)) and large prints (\(y\)) satisfy both conditions:
[tex]\[ \begin{array}{|l|l|l|l|} \hline \multicolumn{4}{|c|}{Combination Options} \\ \hline (45,10) & (35,15) & (30,10) & (40,5) \\ \hline \end{array} \][/tex]
1. Combination \((45, 10)\):
- Check \(15 \cdot 45 + 25 \cdot 10 \geq 700\):
[tex]\[ 675 + 250 = 925 \quad \text{which is} \geq 700 \][/tex]
- Check \(45 > 3 \cdot 10\):
[tex]\[ 45 > 30 \][/tex]
- This combination satisfies both conditions.
2. Combination \((35, 15)\):
- Check \(15 \cdot 35 + 25 \cdot 15 \geq 700\):
[tex]\[ 525 + 375 = 900 \quad \text{which is} \geq 700 \][/tex]
- Check \(35 > 3 \cdot 15\):
[tex]\[ 35 > 45 \][/tex]
- This combination does not satisfy the production condition.
3. Combination \((30, 10)\):
- Check \(15 \cdot 30 + 25 \cdot 10 \geq 700\):
[tex]\[ 450 + 250 = 700 \quad \text{which is} \geq 700 \][/tex]
- Check \(30 > 3 \cdot 10\):
[tex]\[ 30 > 30 \][/tex]
- This combination does not satisfy the production condition.
4. Combination \((40, 5)\):
- Check \(15 \cdot 40 + 25 \cdot 5 \geq 700\):
[tex]\[ 600 + 125 = 725 \quad \text{which is} \geq 700 \][/tex]
- Check \(40 > 3 \cdot 5\):
[tex]\[ 40 > 15 \][/tex]
- This combination satisfies both conditions.
### Conclusion
The valid combinations that satisfy both conditions are:
- \((45, 10)\)
- \((40, 5)\)
Thus, the correct answers are [tex]\((45, 10)\)[/tex] and [tex]\((40, 5)\)[/tex].
### Step 1: Define the Inequalities
We have two conditions to consider:
1. Profit Condition:
The store needs to make at least $700 per day. The profit is calculated based on the small prints (\(x\)) and large prints (\(y\)):
[tex]\[ 15x + 25y \geq 700 \][/tex]
2. Production Condition:
The number of small prints (\(x\)) must be more than three times the number of large prints (\(y\)):
[tex]\[ x > 3y \][/tex]
Given these inequalities, we can identify the correct options from the provided table.
- \(15x + 25y \geq 700\)
- \(x > 3y\)
Now let's match these with the given options:
[tex]\[ \begin{array}{|c|c|c|} \hline \multicolumn{3}{|c|}{Inequality Options} \\ \hline x+y \leq 60 & 15x + 25y < 700 & x > 3y \\ \hline 15x + 25y \geq 700 & y > 3x & x + 3y \geq 60 \\ \hline \end{array} \][/tex]
From this table, the correct inequalities are:
- \(15x + 25y \geq 700\)
- \(x > 3y\)
### Step 2: Verify the Combinations
Now, we need to check which combinations of small prints (\(x\)) and large prints (\(y\)) satisfy both conditions:
[tex]\[ \begin{array}{|l|l|l|l|} \hline \multicolumn{4}{|c|}{Combination Options} \\ \hline (45,10) & (35,15) & (30,10) & (40,5) \\ \hline \end{array} \][/tex]
1. Combination \((45, 10)\):
- Check \(15 \cdot 45 + 25 \cdot 10 \geq 700\):
[tex]\[ 675 + 250 = 925 \quad \text{which is} \geq 700 \][/tex]
- Check \(45 > 3 \cdot 10\):
[tex]\[ 45 > 30 \][/tex]
- This combination satisfies both conditions.
2. Combination \((35, 15)\):
- Check \(15 \cdot 35 + 25 \cdot 15 \geq 700\):
[tex]\[ 525 + 375 = 900 \quad \text{which is} \geq 700 \][/tex]
- Check \(35 > 3 \cdot 15\):
[tex]\[ 35 > 45 \][/tex]
- This combination does not satisfy the production condition.
3. Combination \((30, 10)\):
- Check \(15 \cdot 30 + 25 \cdot 10 \geq 700\):
[tex]\[ 450 + 250 = 700 \quad \text{which is} \geq 700 \][/tex]
- Check \(30 > 3 \cdot 10\):
[tex]\[ 30 > 30 \][/tex]
- This combination does not satisfy the production condition.
4. Combination \((40, 5)\):
- Check \(15 \cdot 40 + 25 \cdot 5 \geq 700\):
[tex]\[ 600 + 125 = 725 \quad \text{which is} \geq 700 \][/tex]
- Check \(40 > 3 \cdot 5\):
[tex]\[ 40 > 15 \][/tex]
- This combination satisfies both conditions.
### Conclusion
The valid combinations that satisfy both conditions are:
- \((45, 10)\)
- \((40, 5)\)
Thus, the correct answers are [tex]\((45, 10)\)[/tex] and [tex]\((40, 5)\)[/tex].
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