I NEED HELP SOON PLEASE

Question

A supply company manufactures copy machines. The unit cost C

 (the cost in dollars to make each copy machine) depends on the number of machines made. If x

 machines are made, then the unit cost is given by the function =Cx+−0.4x2192x33,821

. What is the minimum unit cost?

Do not round your answer.

Answer 1

Assuming the unit cost function is C(x) = 0.4x² - 192x + 33821 then it is a quadratic function meaning the graph formed is in the shape of a parabola. A parabola has a vertex or "tip" which will result in a minimum value if the parabola is upright or a maximum value if the parabola is inverted. In this case, because the coefficient in front of the x² is positive, the parabola is upright so it will have a minimum value. And, we are looking for the minimum unit cost so we want to find that vertex.

The "x" value of a vertex y = ax² + bx + c can be found using the equation x = -b/(2a).

In this case a = 0.4, b = -192 and c = 33821. So the x-value of the vertex is found plugging in those values of "a" and "b" into the equation:

x = -b/(2a)

x = - (-192)/(2•0.4)

x = 192/0.8

x = 240

That tells you that the minimum unit cost occurs when 240 machines are built.

To know what that unit cost is, you must plug in 240 into the original function

C(x) = 0.4x² - 192x + 33821

C(240) = 0.4(240)² - 192(240) + 33821

I'll let you work through the arithmetic to get the final answer

Answer 2

The graph of the function is that of a parabola. The x-coordinate of the vertex of a parabola of the form: f(x) = ax^2 + bx + c can be found using x = -b/2a.

Given the function C(x) = 0.4x^2 - 192x + 33821, the coefficients are a=0.4; b= -192; c=33821.

Then, x = -b/2a = -(-192)/2(0.4) = 240.

The y-coordinate of the vertex will be found by replacing x=240 in the function:

C(240) = 0.4(240)^2 - 192(240) + 33821 = 10781.

The vertex is found at (240,10781).

Since the amplitude (a) is positive, the parabola opens upward, the vertex is a minimum point, and therefore, C(240) = 10781 is the minimum unit cost per copy machine.