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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?

Sagot :

GinnyAnsweridn
Alright, let's tackle this problem step-by-step.

### Step 1: Define the Profit Function

The profit function [tex]\(P(x)\)[/tex] represents the total profit made from selling [tex]\(x\)[/tex] bracelets. It is defined as the difference between the revenue function [tex]\(R(x)\)[/tex] and the cost function [tex]\(C(x)\)[/tex]:

[tex]\[ P(x) = R(x) - C(x) \][/tex]

### Step 2: Given Functions

We are given the cost function [tex]\(C(x)\)[/tex] and the revenue function [tex]\(R(x)\)[/tex]:

[tex]\[ C(x) = 180 + 8x \][/tex]
[tex]\[ R(x) = 20x \][/tex]

### Step 3: Substitute the Given Functions into the Profit Function

Now substitute [tex]\(C(x)\)[/tex] and [tex]\(R(x)\)[/tex] into the profit function:

[tex]\[ P(x) = R(x) - C(x) \][/tex]
[tex]\[ P(x) = 20x - (180 + 8x) \][/tex]

### Step 4: Simplify the Profit Function

Simplify the expression by distributing and combining like terms:

[tex]\[ P(x) = 20x - 180 - 8x \][/tex]
[tex]\[ P(x) = (20x - 8x) - 180 \][/tex]
[tex]\[ P(x) = 12x - 180 \][/tex]

So, the simplified profit function is:

[tex]\[ P(x) = 12x - 180 \][/tex]

### Step 5: Determine the Break-Even Point

The break-even point occurs when the profit [tex]\(P(x)\)[/tex] is zero, meaning the revenue equals the cost. Set [tex]\(P(x)\)[/tex] equal to zero and solve for [tex]\(x\)[/tex]:

[tex]\[ P(x) = 0 \][/tex]
[tex]\[ 12x - 180 = 0 \][/tex]

### Step 6: Solve the Equation

Solve for [tex]\(x\)[/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.
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