To determine how many shirts Carlotta must sell to make a profit, we need to consider her costs and potential earnings. Let’s break down the problem step by step:
1. Understand the Production Cost Equation:
[tex]\[ c = 28 + 35s \][/tex]
Here, [tex]\(c\)[/tex] is the total cost and [tex]\(s\)[/tex] is the number of shirts produced.
- [tex]\(28\)[/tex] is the fixed cost (constant, regardless of how many shirts she produces).
- [tex]\(35s\)[/tex] is the variable cost (dependent on the number of shirts produced).
2. Understand the Money Earned Equation:
[tex]\[ m = 10s \][/tex]
Here, [tex]\(m\)[/tex] is the amount of money earned from selling [tex]\(s\)[/tex] shirts.
3. Determine the Profit Condition:
Carlotta makes a profit if the money earned is greater than the total production cost.
[tex]\[ m > c \][/tex]
Substituting the given equations, we get:
[tex]\[ 10s > 28 + 35s \][/tex]
4. Isolate [tex]\(s\)[/tex] to find the break-even point:
Rearrange the inequality:
[tex]\[ 10s - 35s > 28 \][/tex]
[tex]\[ -25s > 28 \][/tex]
5. Solve for [tex]\(s\)[/tex]:
[tex]\[ s < -\frac{28}{25} \][/tex]
[tex]\[ s < -1.12 \][/tex]
This result is nonsensical because you can’t sell a negative number of shirts, so let's re-examine where we start checking the number of shirts sold to make a profit.
6. Calculate specific values to check profitability:
Let’s test the given shirt counts (2, 3, 4, 5):
- For [tex]\( s = 2 \)[/tex]:
[tex]\[ \text{Total Cost} = 28 + 35 \times 2 = 28 + 70 = 98 \][/tex]
[tex]\[ \text{Money Earned} = 10 \times 2 = 20 \][/tex]
Since [tex]\( 20 < 98 \)[/tex] no profit is made.
- For [tex]\( s = 3 \)[/tex]:
[tex]\[ \text{Total Cost} = 28 + 35 \times 3 = 28 + 105 = 133 \][/tex]
[tex]\[ \text{Money Earned} = 10 \times 3 = 30 \][/tex]
Since [tex]\( 30 < 133 \)[/tex] no profit is made.
- For [tex]\( s = 4 \)[/tex]:
[tex]\[ \text{Total Cost} = 28 + 35 \times 4 = 28 + 140 = 168 \][/tex]
[tex]\[ \text{Money Earned} = 10 \times 4 = 40 \][/tex]
Since [tex]\( 40 < 168 \)[/tex] no profit is made.
- For [tex]\( s = 5 \)[/tex]:
[tex]\[ \text{Total Cost} = 28 + 35 \times 5 = 28 + 175 = 203 \][/tex]
[tex]\[ \text{Money Earned} = 10 \times 5 = 50 \][/tex]
Since [tex]\( 50 < 203 \)[/tex] no profit is made.
From testing the above values, it is clear that no profit is made with 2, 3, 4, or 5 shirts.
Conclusion:
No profit can be made with 2 to 5 shirts. Carlotta must sell more than 5 shirts to make a profit.
