earlakey
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For a certain company, the cost for producing x items is 60x+300 and the revenue for selling x items is 100x−0.5x2.



The profit that the company makes is how much it takes in (revenue) minus how much it spends (cost). In economic models, one typically assumes that a company wants to maximize its profit, or at least wants to make a profit!



Part a: Set up an expression for the profit from producing and selling x items. We assume that the company sells all of the items that it produces. (Hint: it is a quadratic polynomial.)



Part b: Find two values of x that will create a profit of $50 .

Respuesta :

MrRoyal

The profit function of the company is the difference between the cost and the revenue functions of the company

  • The expression of the profit function is [tex]- 0.5x^2 + 40x - 300[/tex]
  • The two values of x that creates a profit of $50 are x = 10 and x = 70

How to set up the profit function

The given parameters are:

C(x) = 60x + 300 --- the cost function

R(x) = 100x - 0.5x^2 ---- the revenue function

The profit function is calculated as:

P(x) = R(x) - C(x)

So, we have:

[tex]P(x) = 100x - 0.5x^2 - 60x - 300[/tex]

Rewrite as:

[tex]P(x) = - 0.5x^2 + 100x - 60x - 300[/tex]

[tex]P(x) = - 0.5x^2 + 40x - 300[/tex]

This means that the expression of the profit function is [tex]- 0.5x^2 + 40x - 300[/tex]

The two values of x that creates a profit of $50

This means that:

P(x) = 50

So, we have:

[tex]- 0.5x^2 + 40x - 300 = 50[/tex]

Subtract 50 from both sides

[tex]- 0.5x^2 + 40x - 350 = 0[/tex]

Multiply through by -2

[tex]x^2 - 80x + 700 = 0[/tex]

Expand

[tex]x^2 - 70x - 10x + 700 = 0[/tex]

Factorize

[tex]x(x - 70) - 10(x - 70) = 0[/tex]

Factor out x - 70

[tex](x - 10)(x - 70) = 0[/tex]

Solve for x

x = 10 or x = 70

Hence, the two values of x that creates a profit of $50 are x = 10 and x = 70

Read more about profit functions at:

https://brainly.com/question/19104371

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