Bartech,Inc.is a firm operating in a competitive market.The manager of Bartech forecasts product price to be $28 in 2021.Bartech's average variable cost function is estimated to be $A V C = 10 - 0.003 Q + 0.0000005 Q ^ { 2 }$ Bartech expects to face fixed costs of $12,000 in 2021.What is the minimum average variable cost?

Question

Quizplus · Accepted Answer

The minimum average variable cost (AVC) can be found by taking the first derivative of the AVC function with respect to Q and setting it equal to zero to find the quantity (Q) that minimizes AVC. The AVC function is $AVC = 10 - 0.003Q + 0.0000005Q^2$. The first derivative of AVC with respect to Q is $-0.003 + 0.000001Q$. Setting this equal to zero gives $0.000001Q = 0.003$, so $Q = 3000$. Substituting Q = 3000 back into the original AVC function gives $AVC = 10 - 0.003(3000) + 0.0000005(3000)^2 = 10 - 9 + 4.5 = 5.50$.

Bartech,Inc AVC=10−0.003Q+0.0000005Q2A V C = 10 - 0.003 Q + 0.0000005 Q ^ { 2 }AVC=10−0.003Q+0.0000005Q2

Bartech,Inc $A V C = 10 - 0.003 Q + 0.0000005 Q ^ { 2 }$