SOLUTION: Hi there, I have some of the answers to this following question, but I'm needing help in some of the other areas, I would really appreciate it! The cost, in dollars, for a compa

Question 1039158: Hi there, I have some of the answers to this following question, but I'm needing help in some of the other areas, I would really appreciate it!
The cost, in dollars, for a company to produce x widgets is given by C(x) = 3600 + 5x for
x greater than or equal to 0, and the price-demand function, in dollars per widget, is p(x) = 45 - 0.04x for 0 less than or equal to x less than or equal to 1125.

(a) The profit function is a quadratic function and so its graph is a parabola.
Does the parabola open up or down? __down____
(b) Find the vertex of the profit function P(x) using algebra. Show algebraic work.
(500, 6500)

(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.
I would really appreciate help on c, d and e. Thank you!

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
*******************************************************************************
Part c
:
The parabola which represents the profit function curves downward so use vertex coordinates
:
The maximum profit is 6500 for 500 widgets produced
:
*******************************************************************************
Part d
:
p(500) = 45 - 0.04(500) = 25
*******************************************************************************
Part e
:
The marginal cost is the slope of the cost curve
C(x) = 3600 + 5x
The marginal cost is 5
:
revenue function is (45 - 0.04x)x= 45x - 0.04x^2
marginal revenue is the first derivative
45 - 0.08x
:
profit is revenue - cost
45x - 0.04x^2 - (3600 +5x)
P(x) = -0.04x^2 + 40x -3600
:
max profit is found by taking the first derivative of P(x) and setting it = 0, then solve for x and substitute for x in P(x)
-0.08x + 40 = 0
x = 500
:
The break-even points are where the graph of P(x) crosses the x-axis
:

:

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