To solve this problem, we need to determine the profit function [tex]\( P(x) \)[/tex] from the given cost and revenue functions, and then find the break-even point where the company makes no profit and no loss.
### Step 1: Writing the Profit Function
The profit function [tex]\( P(x) \)[/tex] is the revenue function [tex]\( R(x) \)[/tex] minus the cost function [tex]\( C(x) \)[/tex]:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Given:
[tex]\[ R(x) = 20x \][/tex]
[tex]\[ C(x) = 180 + 8x \][/tex]
Substituting these into the profit function:
[tex]\[ P(x) = 20x - (180 + 8x) \][/tex]
### Step 2: Simplifying the Profit Function
Simplify the expression by 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 3: Finding the Break-Even Point
To find the break-even point, we need to determine when the profit [tex]\( P(x) \)[/tex] is zero:
[tex]\[ P(x) = 0 \][/tex]
Set the profit function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 12x - 180 = 0 \][/tex]
Add 180 to both sides:
[tex]\[ 12x = 180 \][/tex]
Divide both sides by 12:
[tex]\[ x = \frac{180}{12} \][/tex]
[tex]\[ x = 15 \][/tex]
### Conclusion
To break even, the company must sell 15 bracelets.
