How many tractors should be produced to incur minimum cost?

Question

The total cost of producing a type of tractor is given by C(x) = 18000 − 60x + 0.05x2, where x is the number of tractors produced. How many tractors should be produced to incur minimum cost?

Answer 1

Bill K's answer is right.  And it is a quick way of figuring out this problem.

However, if you're not working with differential equations yet you can work

this by using the quadratic equation:

 

C(x) = 18000 - 60x + 0.05x²

 

Let's rearrange the equation in order of descending exponents:

 

C(x) = .05x² -60x + 18000

This is a quadratic equation of the type ax²+bx+c graphing as

a parabola opening upward.  The minimum cost will be found at

the vertex since that will be the lowest point on the graph.

 

The vertex of the parabola is (-b/2a, C(-b/2a))

a = .05, b = -60

 

-b/2a = 60/.1 = 600

The minimum cost occurs when we make 600 tractors.

 

 

Answer 2

Minimum cost can be found by differentiating cost function and setting the result to zero

C' = -60 + 0.1x = 0

0.1x = 60

x = 600 tractors