Please help The cost, in dollars, for a company to produce x widgets is given by

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The cost, in dollars, for a company to produce x widgets is given by C(x) = 4200 + 5.00x for x greaterthanorequalto 0, and the price-demand function, in dollars per widget, is p(x) = 38 - 0.02x for 0 lessthanorequalto x lessthanorequalto 1900. In Quiz 2, problem #10, we saw that the profit function for this scenario is P(x) = 0.02x^2 + 33.00x - 4200. (a) The profit function is a quadratic function and so its graph is a parabola. Does the parabola open up or down? _____ (b) Find the vertex of the profit function P(x) using algebra. Show algebraic work. (c) State the maximum profit and the number of widgets which yield that maximum profit: The maximum profit is _____ when _____ widgets are produced and sold. (d) Determine the price to charge per widget in order to maximize profit. (e) Find and interpret the break-even points. Show algebraic work.

Explanation / Answer

p(x) = -0.02x^2 + 33.00x - 4200

since the leading coefficient of profit function is negative hence the parabola opens downwards

b) vertex of profit function is given by the formula

x = -b/2a

x = -33.00 / 2* -.02

x = 825

y = -0.02(825)^2 + 33.00(825) - 4200

y = 9412.5

vertex is ( 825 , 9412.5 )

c) maximum profit is 9412.5 when 825 widgets are produced and sold

d) price per widget = 38 - 0.02 ( 825 ) = 21.5

e) breakeven points are found by setting P(x) = 0

-0.02x^2 + 33.00x - 4200 = 0

x = { -33 + - sqrt ( 33^2 - 4*(-.02)(-4200) } / 2* -.02

x = 138.98 , 1511.02

hence, 2 breakeven points are 138.98 , 1511.02

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