To determine how many shirts Carlotta needs to sell to make a profit, we need to compare her production costs with her earnings from selling the shirts.
Given:
- Production Cost: [tex]\( c = 28 + 3s \)[/tex], where [tex]\( c \)[/tex] represents the total cost and [tex]\( s \)[/tex] represents the number of shirts.
- Money Earned: [tex]\( m = 10s \)[/tex], where [tex]\( m \)[/tex] represents the total money earned from selling the shirts and [tex]\( s \)[/tex] represents the number of shirts sold.
The goal is to find the smallest number [tex]\( s \)[/tex] such that the money earned ([tex]\( m \)[/tex]) is greater than the production cost ([tex]\( c \)[/tex]):
[tex]\[ m > c \][/tex]
Substitute the given formulas into the inequality:
[tex]\[ 10s > 28 + 3s \][/tex]
Now, solve this inequality for [tex]\( s \)[/tex]:
1. Subtract [tex]\( 3s \)[/tex] from both sides to isolate the term involving [tex]\( s \)[/tex] on one side:
[tex]\[ 10s - 3s > 28 \][/tex]
2. Simplify the left side:
[tex]\[ 7s > 28 \][/tex]
3. Divide both sides by 7 to solve for [tex]\( s \)[/tex]:
[tex]\[ s > \frac{28}{7} \][/tex]
[tex]\[ s > 4 \][/tex]
This inequality tells us that [tex]\( s \)[/tex] must be greater than 4. The smallest integer greater than 4 is 5. Therefore, Carlotta must sell at least 5 shirts to make a profit.
Thus, the answer is:
5 shirts
