The cost function for production of a commodity is C(x) = 352 + 30x - 0.05x^2 +

Question

The cost function for production of a commodity is C(x) = 352 + 30x - 0.05x^2 + 0.0009x^3. Find C'(100). Interpret C'(100). This is the rate at which costs are increasing with respect to the production level when x = 100. This is the number of items that must be produced before the costs reach 100. This is the cost of making 100 items. This is the amount of time, in minutes, it takes to produce 100 items. This is the rate at which the production level is decreasing with respect to the cost when x = 100. Find the actual cost of producing the 101st item. (Round your answer to the nearest cent.)

Explanation / Answer

part a

C'(X) = 30- 0.1x + 0.0027x2

C'(100)= 30-0.1*100 + 0.0027*1002 = 47

this is the rate at which cost are increasing with respect to production level when x=100

C(100) = 352+30*100-0.05*1002 + 0.0009*1003 = 3752

C(101) = 352+30*101-0.05*1012+0.0009*1013 = 3799.2209

so cost of 101st = C(101)-C(100) = 47.2209

second method

as we know rate of production after 100

C'(100) is 47 so 101st will be 47

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