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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
To determine the profit made from selling [tex]\(x\)[/tex] charm bracelets, we need to define the profit function [tex]\(P(x)\)[/tex]. Profit is calculated as the difference between revenue and cost. Given the cost function [tex]\(C(x)\)[/tex] and the revenue function [tex]\(R(x)\)[/tex], we can write:

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

The profit function [tex]\(P(x)\)[/tex] is then:

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

Substituting the given expressions for [tex]\(R(x)\)[/tex] and [tex]\(C(x)\)[/tex]:

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

Simplifying this expression, we get:

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

So, the simplified profit function is:

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

To find the break-even point, we need to determine the value of [tex]\(x\)[/tex] for which the profit [tex]\(P(x)\)[/tex] is zero. Setting [tex]\(P(x) = 0\)[/tex]:

[tex]\[ 0 = 12x - 180 \][/tex]

Solving for [tex]\(x\)[/tex]:

[tex]\[ 12x = 180 \][/tex]
[tex]\[ x = \frac{180}{12} \][/tex]
[tex]\[ x = 15 \][/tex]

Therefore, the company must sell 15 bracelets to break even.
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