The cost function, in dollars, of a company that manufactures food processors is given by C(x)=200+7/x+x^2/7, where x is the number of food processors manufactured.

Question

1. Find the marginal cost function.
2. Find the marginal cost of manufacturing 12 food processors.
3. Find the actual cost of manufacturing the thirteenth food processor.

any help would be amazing

Answer 1

Total Cost =C(x) = 200 + 7/x + x^2/7

Marginal Cost = the derivative of C(x) = C'(x) = -7/x^2 + (2/7)x

C'(12) = - 7/(144) + (2/7)(12)

= ,-0486111... + 3.428571429... = about 3.38 = $3.38

Marginal Cost of 12th = $3.38

C(12) = 200 +7/12 + 144/7

C(13) = 200 + 7/13 + 169/7

C(13)-C(12) = (169-144)/7 + (7/13- 7/12) = 125/7 + 7/13 - 7/12

put the fractions over a common denominator: 7x13x12

= (12x13x25 +7x7x12 - 7x7x13)/(7x12x13)

= (3900 - 637 + 588)/(7x12x13)= 3851/7x12x13 =

= 3.53

actual cost of 13th = $3.53

C'(13) = -7/169 + (2/7)13 = $3.67

C'(12) = $3.38

average of marginal costs for 12th and 13th

(C'(13)+C'(12))/2 = (3.67+3.38)/2 = $3.53 = the same as the actual cost of the 13th