Certainly! Let's go through the problem step by step to find the solution.
### Step 1: Define the Cost Function
The cost function [tex]\( C(x) \)[/tex] represents the total cost of producing [tex]\( x \)[/tex] bracelets. It is given by:
[tex]\[ C(x) = 180 + 8x \][/tex]
### Step 2: Define the Revenue Function
The revenue function [tex]\( R(x) \)[/tex] represents the total revenue from selling [tex]\( x \)[/tex] bracelets. It is given by:
[tex]\[ R(x) = 20x \][/tex]
### Step 3: Define and Simplify the Profit Function
The profit function [tex]\( P(x) \)[/tex] is defined as the difference between the revenue and the cost. Therefore, we get:
[tex]\[ P(x) = R(x) - C(x) \][/tex]
Substitute the given functions into this equation:
[tex]\[
P(x) = 20x - (180 + 8x)
\][/tex]
Simplify the equation:
[tex]\[
P(x) = 20x - 180 - 8x
\][/tex]
[tex]\[
P(x) = 12x - 180
\][/tex]
So the profit function [tex]\( P(x) \)[/tex] is:
[tex]\[ P(x) = 12x - 180 \][/tex]
### Step 4: Determine the Break-Even Point
To break even, the profit must be zero. Therefore, we set [tex]\( P(x) \)[/tex] 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
The company must sell 15 bracelets to break even.
