Alright, let's tackle this problem step-by-step.
### Step 1: Define the Profit Function
The profit function [tex]\(P(x)\)[/tex] represents the total profit made from selling [tex]\(x\)[/tex] bracelets. It is defined as the difference between the revenue function [tex]\(R(x)\)[/tex] and the cost function [tex]\(C(x)\)[/tex]:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
### Step 2: Given Functions
We are given the cost function [tex]\(C(x)\)[/tex] and the revenue function [tex]\(R(x)\)[/tex]:
[tex]\[ C(x) = 180 + 8x \][/tex]
[tex]\[ R(x) = 20x \][/tex]
### Step 3: Substitute the Given Functions into the Profit Function
Now substitute [tex]\(C(x)\)[/tex] and [tex]\(R(x)\)[/tex] into the profit function:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
[tex]\[ P(x) = 20x - (180 + 8x) \][/tex]
### Step 4: Simplify the Profit Function
Simplify the expression by distributing and combining like terms:
[tex]\[ P(x) = 20x - 180 - 8x \][/tex]
[tex]\[ P(x) = (20x - 8x) - 180 \][/tex]
[tex]\[ P(x) = 12x - 180 \][/tex]
So, the simplified profit function is:
[tex]\[ P(x) = 12x - 180 \][/tex]
### Step 5: Determine the Break-Even Point
The break-even point occurs when the profit [tex]\(P(x)\)[/tex] is zero, meaning the revenue equals the cost. Set [tex]\(P(x)\)[/tex] equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ P(x) = 0 \][/tex]
[tex]\[ 12x - 180 = 0 \][/tex]
### Step 6: Solve the Equation
Solve for [tex]\(x\)[/tex]:
[tex]\[ 12x = 180 \][/tex]
[tex]\[ x = \frac{180}{12} \][/tex]
[tex]\[ x = 15 \][/tex]
### Conclusion
The company must sell 15 bracelets to break even.
