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

The minimum unit cost is 5377

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=-(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=-(-400)/(2(1))`

`x=200\ cars`

So the minimum unit cost is:

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

`C(200) = 5377`

Answer 2

5377

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