To determine the break-even point, we need to find the number of shirts [tex]\( n \)[/tex] where the total revenue equals the total cost. The revenue function and cost function are given as:
[tex]\[ r = 15n \][/tex]
[tex]\[ C = 9n + 450 \][/tex]
At the break-even point, revenue equals cost. Therefore, we set the two functions equal to each other:
[tex]\[ 15n = 9n + 450 \][/tex]
To solve for [tex]\( n \)[/tex], follow these steps:
1. Subtract [tex]\( 9n \)[/tex] from both sides of the equation to isolate the terms involving [tex]\( n \)[/tex] on one side:
[tex]\[ 15n - 9n = 450 \][/tex]
2. Simplify the left-hand side:
[tex]\[ 6n = 450 \][/tex]
3. Divide both sides of the equation by 6 to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \frac{450}{6} \][/tex]
4. Calculate the value:
[tex]\[ n = 75 \][/tex]
Thus, the break-even point is when Amanda sells 75 T-shirts. Therefore, the correct answer is:
B. [tex]\( n = 75 \)[/tex]
