To write and simplify a function [tex]\( P(x) \)[/tex] that represents the profit made from selling [tex]\( x \)[/tex] bracelets, we start by defining the cost and revenue functions given:
- The cost function [tex]\( C(x) \)[/tex] is given as:
[tex]\[
C(x) = 180 + 8x
\][/tex]
- The revenue function [tex]\( R(x) \)[/tex] is given as:
[tex]\[
R(x) = 20x
\][/tex]
The profit function [tex]\( P(x) \)[/tex] is defined as the difference between the revenue and the cost:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
Substitute the given functions for [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex] into this equation:
[tex]\[
P(x) = 20x - (180 + 8x)
\][/tex]
Now, simplify the profit function [tex]\( P(x) \)[/tex]:
[tex]\[
P(x) = 20x - 180 - 8x = 20x - 8x - 180 = 12x - 180
\][/tex]
So, the simplified profit function is:
[tex]\[
P(x) = 12x - 180
\][/tex]
Next, to find the number of bracelets the company must sell to break even, we need to determine when the profit function equals 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]\[
x = \frac{180}{12} = 15
\][/tex]
Therefore, the company must sell 15 bracelets to break even.
