Let's break down the solution in detail, following each step methodically to understand the breakeven point:
1. Understanding Profit Function:
We start with the profit function, defined as the difference between revenue and cost. Given:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
2. Given Functions:
We are provided with the following expressions:
[tex]\[
R(x) = 20x
\][/tex]
[tex]\[
C(x) = 180 + 8x
\][/tex]
3. Construct the Profit Function:
Subtract the cost function [tex]\(C(x)\)[/tex] from the revenue function [tex]\(R(x)\)[/tex]:
[tex]\[
P(x) = 20x - (180 + 8x)
\][/tex]
Simplifying the expression inside the parentheses:
[tex]\[
P(x) = 20x - 180 - 8x
\][/tex]
4. Combine Like Terms:
Combine the terms with [tex]\(x\)[/tex] and the constants:
[tex]\[
P(x) = 12x - 180
\][/tex]
5. Find the Breakeven Point:
The breakeven point is where the profit [tex]\(P(x)\)[/tex] equals zero:
[tex]\[
0 = 12x - 180
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
To find [tex]\(x\)[/tex], first isolate the term with [tex]\(x\)[/tex]:
[tex]\[
180 = 12x
\][/tex]
Then, solve for [tex]\(x\)[/tex] by dividing both sides by 12:
[tex]\[
x = \frac{180}{12} = 15
\][/tex]
7. Conclusion:
Therefore, the breakeven point occurs when 15 bracelets are sold. This means the company needs to sell 15 bracelets to cover its costs and start making a profit.
So, the number of bracelets that must be sold to break even is 15.
Therefore, the breakeven point is when:
[tex]\[
(x_{\text{breakeven}}, 0) = (15, 0)
\][/tex]
In conclusion, based on the calculations, the company breaks even after selling 15 bracelets.
