can someone please help with this world problem.

Question

I am confused on this question and not sure if i am supposed to be using the first equation or the second. Can someone please help me understand this equation because I am so lost and dont know how to begin to solve this.

 

The cost, in dollars, for a company to produce x widgets is given by C(x) = 4700 + 5.20x for
x ≥ 0, and the price-demand function, in dollars per widget, is p(x) = 40 - 0.02x for 0 ≤ x ≤ 2000.

In Quiz 2, problem #10, we saw that the profit function for this scenario is
P(x) = - 0.02x² + 34.80x - 4700.

(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.


(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) How many widgets can be produced and sold in order to earn at least $7700 profit?
Hint: Start by solving the equation P(x) = 7700.
State your conclusion clearly in a sentence.

Answer 1

The leading coefficient of the Profit function is -0.02, so it opens downward.

 

The axis of symmetry is x = -b/2a = -(34.80)/2(-0.02) = 870

 

The value of the function at 870, the local maximum, is

= -0.02(870)² + 34.80(870) - 4700

= -0.02(756900) + 30276 - 4700

=  -15138 + 25576

= 10438

 

So the vertex is (870, 10438)

 

The cost t the company to produce 870 widgets is 

C(870) = 4700 + 5.20(870) = 4700 + 4524 = 9224

 

So, the cost of the widgets plus the profit must be equal to the total sales, whih divided by the number of widgets reveals their individual price.

(10438 + 9224)/870 = 19662/870 = $22.60

 

P(x) = 7700

- 0.02x² + 34.80x - 4700 = 7700

-0.02x² + 34.80x -12400 = 0

 

x = {-34.80 ± √[(34.80)² - 4(-0.02)(-12400)]}/2(-0.02)

x = [-34.80 ± √(1211.04 - 992)]/(-0.04)

x = (-34.80 ± √219.04)/(-0.04)

x = (-34.80 ± 14.8)/(-0.04)

x = 870 ± 370

so, $7700 in profits will be earned at either 500 widgets or 1240 widgets

 

Therefore at least 500 widgets must be produced to achieve profits of at least $7700, but you should take care not to produce more than 1240 widgets.