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A company makes and sells charm bracelets. The cost of producing [tex]\( x \)[/tex] bracelets is represented by the function [tex]\( C(x) = 180 + 8x \)[/tex]. The revenue earned from selling [tex]\( x \)[/tex] bracelets is represented by the function [tex]\( R(x) = 20x \)[/tex].

1. Write and simplify a function [tex]\( P(x) \)[/tex] that represents the profit made from selling [tex]\( x \)[/tex] bracelets.
2. How many bracelets must the company sell to break even?



Answer :

GinnyAnswer
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.

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