find the production level that will maximize profit.If  C(x) = 15000 + 600x − 2.8^2+ 0.004x^3 is the cost function and  p(x) = 4200 − 7x  is the demand function, find the production level...

find the production level that will maximize profit.

If  C(x) = 15000 + 600x − 2.8^2+ 0.004x^3 is the cost function and  p(x) = 4200 − 7x  is the demand function, find the production level that will maximize profit. (Hint: If the profit is maximized, then the marginal revenue equals the marginal cost.)

 

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember that the marginal cost equals marginal revenue means that C'(x) = p'(x), hence you should find the derivative of C(x) and p(x) with respect to x such that:

`C'(x) = 600 - 5.6x + 0.012x^2`

`p'(x) = -7`

You need to set the equations of C'(x) and p'(x) equal such that:

`600 - 5.6x + 0.012x^2 = -7`

`0.012x^2 - 5.6x + 607 = 0`

Using quadratic formula yields:

`x_(1,2) = (5.6+-sqrt(31.36 - 29.136))/0.024`

`x_(1,2) = (5.6+-1.4)/0.024`

`x_1 = 7/0.024 =gt x_1 = 291.6`

`x_2 = 175`

Hence, evaluating the values of x for the marginal cost equals marginal revenue yields `x_1~~ 292`  and `x_2 = 175` .

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