The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the ...

Question

The Marginal cost of a product can be thought of as the cost of producing one additional unit of  output. For example, if the marginal cost of producing  the 50th product is $6.20, it cost $6.20 to increase production from 49 to 50 units of output. Suppose the marginal cost C  (in dollars) to produce x thousand mp3 players is given by the function C(x)=x^2-100x+8300.  (a). How many players should be produced to minimize the marginal cost? and (b). What is the minimum cost??

Answer 1

 If you take Calculus marginal  is the derivative, the value of change at a point(margin).

 

  C( X) = X² - 100X + 8300

 

  a) This is a quadratic function, the minimum value is @ - b/ 2a:

    

      - ( -100) /2 = 50

  b) 

          C( 50) = (50) ² - 100(50) + 8300 = 5800

 

       Another way, using Calculus:

 

       a)  dC/dx =  2x - 100 =50

 

 

Answer 2

The Marginal cost is found by taking the derivative of the cost equation.

 

C(x) = x^2 - 100x + 8300

 

C'(x) = 2x - 100

 

This curve will have a minimum since it's a parabola that is concave up. 

 

2x - 100 = 0

 

2x = 100

 

x = 50

 

50,000 mp3 players should be made to minimize the marginal cost

 

The minimum cost will be 50^2 - 100*50 + 8300 = $5800

Answer 3

Hey Kevin -- at a minimum, the slope of the trough is zero ... take dC(X)/dX =0 ...

 

a) 2x -100 =0 ... x= 50 ==> produce 50k mp3 players

b) C(50) = 50*50 less 5k plus 8300 = 5800 min cost at x= 50 ... Best wishes, sir  :)