Let's start by defining the functions given in the problem:
The cost function [tex]\( C(x) \)[/tex] represents the total cost to produce [tex]\( x \)[/tex] bracelets:
[tex]\[ C(x) = 180 + 8x \][/tex]
The revenue function [tex]\( R(x) \)[/tex] represents the total revenue earned from selling [tex]\( x \)[/tex] bracelets:
[tex]\[ R(x) = 20x \][/tex]
The profit function [tex]\( P(x) \)[/tex] is defined as the difference between the revenue [tex]\( R(x) \)[/tex] and the cost [tex]\( C(x) \)[/tex]:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given functions [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex] into the profit function:
[tex]\[ P(x) = 20x - (180 + 8x) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ P(x) = 20x - 180 - 8x \][/tex]
Combine like terms:
[tex]\[ P(x) = 12x - 180 \][/tex]
So, the simplified profit function is:
[tex]\[ P(x) = 12x - 180 \][/tex]
Next, we need to determine the break-even point, which is the number of bracelets [tex]\( x \)[/tex] that must be sold to make the profit equal to zero.
To find the break-even point, set [tex]\( P(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ 12x - 180 = 0 \][/tex]
Solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex] on one side of the equation:
[tex]\[ 12x = 180 \][/tex]
[tex]\[ x = \frac{180}{12} \][/tex]
[tex]\[ x = 15 \][/tex]
Therefore, the company must sell 15 bracelets to break even.
