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Homework answers / question archive / For the given cost function C(x)=250√x+x227000C(x)=250x+x227000 find A) the production level that will minimize the average cost
For the given cost function C(x)=250√x+x227000C(x)=250x+x227000 find
A) the production level that will minimize the average cost.
B) the minimal average cost.
The cost function given is:
C(x)=250x1/2+x227000C(x)=250x1/2+x227000
A)
We first need to find the average cost function. As the average cost is the total cost divided by the number of units produced, the average cost function will be:
AC=250x1/2+x227000x=250x−1/2+x27000AC=250x1/2+x227000x=250x−1/2+x27000
The production level that minimizes the average cost can be found by equating the AC function's derivative to zero and solving for x.
AC′=−225x−1.5+127000AC′=0⇒225x−1.5=127000x=33293.86AC′=−225x−1.5+127000AC′=0⇒225x−1.5=127000x=33293.86
Thus, the average cost is minimized at 33294 units.
B)
The minimum average cost is
AC=250∗33294−1/2+3329427000=2.6
