the production costs per week for generating x

Question

The production costs per week for generating x widgets is given by,

C ( x ) = 500 + 350 x - 0.09 x² ,         0 ≤ x ≤ 1000

Answer following questions.

 (a) What is the cost to generate the 301st widget?

 (b) At x = 300 what is the rate of change of the cost?

Solution

 (a) We can't only compute C (301) as that is the cost of generating 301 widgets whereas we are looking for the actual cost of generating the 301st widget.  In other terms, what we're looking for here is,

C (301) - C (300) = 97, 695.91 - 97, 400.00 + 295.91

Thus, the cost of generating the 301st widget is $295.91.

 (b) In this part all we have to do is get the derivative and then calculate C′ (300) .

                    C′ ( x ) = 350 - 0.18x     ⇒         C′ (300) = 296.00