To determine the profit made from selling [tex]\(x\)[/tex] charm bracelets, we need to define the profit function [tex]\(P(x)\)[/tex]. Profit is calculated as the difference between revenue and cost. Given the cost function [tex]\(C(x)\)[/tex] and the revenue function [tex]\(R(x)\)[/tex], we can write:
[tex]\[ C(x) = 180 + 8x \][/tex]
[tex]\[ R(x) = 20x \][/tex]
The profit function [tex]\(P(x)\)[/tex] is then:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substituting the given expressions for [tex]\(R(x)\)[/tex] and [tex]\(C(x)\)[/tex]:
[tex]\[ P(x) = 20x - (180 + 8x) \][/tex]
Simplifying this expression, we get:
[tex]\[ P(x) = 20x - 180 - 8x \][/tex]
[tex]\[ P(x) = 12x - 180 \][/tex]
So, the simplified profit function is:
[tex]\[ P(x) = 12x - 180 \][/tex]
To find the break-even point, we need to determine the value of [tex]\(x\)[/tex] for which the profit [tex]\(P(x)\)[/tex] is zero. Setting [tex]\(P(x) = 0\)[/tex]:
[tex]\[ 0 = 12x - 180 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ 12x = 180 \][/tex]
[tex]\[ x = \frac{180}{12} \][/tex]
[tex]\[ x = 15 \][/tex]
Therefore, the company must sell 15 bracelets to break even.