Question

A vehicle factory manufactures cars. The unit cost C (the cost in dollars to make each car) depends on the number of cars made. If x cars are made, then the unit cost is given by the function C(x) = x^2 - 400x + 45,377 . What is the minimum unit cost?
Do not round your answer.

Answer 1

Answer:

5377

Step-by-step explanation:

C(x) = x^2 - 400x + 45,377

To find the location of the minimum, we take the derivative of the function

We know that is a minimum since the parabola opens upward

dC/dx = 2x - 400

We set that equal to zero

2x-400 =0

Solving for x

2x-400+400=400

2x=400

Dividing by2

2x/2=400/2

x=200

The location of the minimum is at x=200

The value is found by substituting x back into the equation

C(200) = (200)^2 - 400(200) + 45,377

           =40000 - 80000+45377

            =5377

Answer 2

Answer:

The minimum unit cost is 5377

Step-by-step explanation:

Note that we have a cudratic function of negative principal coefficient.

The minimum value reached by this function is found in its vertex.

For a quadratic function of the form

$ax ^ 2 + bx + c$

the x coordinate of the vertex is given by the following expression

$x=-\frac{b}{2a}$

In this case the function is:

$C(x) = x^2 - 400x + 45,377$

So:

$a=1\\b=-400\\c=45,377$

Then the x coordinate of the vertex is:

$x=-\frac{-400}{2(1)}$

$x=200\ cars$

So the minimum unit cost is:

$C(200) = (200)^2 - 400(200) + 45,377$

$C(200) = 5377$