13. (18 pts) The cost, in dollars, for a company to produce x widgets is given b

Question

13. (18 pts)

The cost, in dollars, for a company to produce x widgets is given by C(x) = 5000 + 3.00x for

x ³ 0, and the price-demand function, in dollars per widget, is p(x) = 28 - 0.02x for 0 £ x £ 1400.

In Quiz 1, problem #9(d), we saw that the profit function for this scenario is

              P(x) = ______________________________. (fill in with the profit function given in the Quiz 1 solutions)

(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

a) first finding revenue function

R(x) = x * p(x) = x (  28 - 0.02x) = 28x - 0.02x^2

profit function = revenue - cost =  28x - 0.02x^2 - (5000 + 3.00x )

P(x) = -0.02x^2 + 25x - 5000

since the leading coeffcient is negative hence parabola opens downwards

b) vertex of parabola is given by

x = -b/2a = -25 / 2 (-0.02) = 625

y = -0.02(625)^2 + 25(625) - 5000 = 2812.5

c) maximum profit is 2812.5 at x = 625 widgets

e) breakeven point occurs when p(x) crosses the x axis

-0.02x^2 + 25x - 5000 = 0

breakeven points are x = 250 , 1000

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.