Sure! Let’s go through the problem step by step.
### 1. Define the Cost Function [tex]\( C(x) \)[/tex]
The cost function [tex]\( C(x) \)[/tex] that represents the cost of producing [tex]\( x \)[/tex] bracelets is given by:
[tex]\[ C(x) = 180 + 8x \][/tex]
### 2. Define the Revenue Function [tex]\( R(x) \)[/tex]
The revenue function [tex]\( R(x) \)[/tex] that represents the revenue earned from selling [tex]\( x \)[/tex] bracelets is given by:
[tex]\[ R(x) = 20x \][/tex]
### 3. Define the Profit Function [tex]\( P(x) \)[/tex]
The profit function [tex]\( P(x) \)[/tex] represents the profit made from selling [tex]\( x \)[/tex] bracelets and is defined as the revenue minus the cost:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
### 4. Substitute the Given Functions into the Profit Function
Substitute [tex]\( R(x) = 20x \)[/tex] and [tex]\( C(x) = 180 + 8x \)[/tex] into the profit function:
[tex]\[ P(x) = 20x - (180 + 8x) \][/tex]
### 5. Simplify the Profit Function
Simplify the expression for [tex]\( P(x) \)[/tex]:
[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]
### 6. Determine the Break-Even Point
To find the break-even point, we need to determine when the profit [tex]\( P(x) \)[/tex] is zero. This occurs when:
[tex]\[ P(x) = 0 \][/tex]
Substitute the profit function into this equation:
[tex]\[ 12x - 180 = 0 \][/tex]
### 7. Solve for [tex]\( x \)[/tex]
Solve the equation for [tex]\( x \)[/tex]:
[tex]\[ 12x - 180 = 0 \][/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.
