A manufacturing enterprise operating in the short run incurs a Total Fixed Cost (\text{TFC… — JAMB UTME

A manufacturing enterprise operating in the short run incurs a Total Fixed Cost ( $\text{TFC}$ ) of $\text{₦}250$ . Its Average Variable Cost ( $\text{AVC}$ ) function is expressed as $\text{AVC} = 3Q^2 - 24Q + 75$ , where $Q$ represents the output level in units. What is the firm's Average Total Cost ( $\text{ATC}$ ) in Naira ( $\text{₦}$ ) at the output level where Average Variable Cost ( $\text{AVC}$ ) reaches its minimum?

Answer: 89.5 Naira

Answer

The firm's Average Total Cost (ATC) at the output level where Average Variable Cost (AVC) is minimized is 89.5 Naira.

To find the Average Total Cost at the output level of minimum Average Variable Cost, first minimize AVC by taking its derivative with respect to output Q and setting it to zero: 6Q - 24 = 0, yielding Q = 4 units. Evaluating AVC at Q = 4 gives AVC = 27 Naira. Next, compute Average Fixed Cost (AFC) at Q = 4 using AFC = TFC / Q = 250 / 4 = 62.5 Naira. Adding AFC and AVC together gives ATC = 62.5 + 27 = 89.5 Naira.

Step-by-Step Solution

Determine the output level Q that minimizes Average Variable Cost (AVC).

Q = 4 units

Setting the derivative of the AVC function with respect to Q equal to zero (6Q - 24 = 0) yields Q = 4.

Compute Average Variable Cost (AVC) at Q = 4.

AVC = 27 Naira

Substituting Q = 4 into AVC = 3(4)^2 - 24(4) + 75 gives 48 - 96 + 75 = 27.

Compute Average Fixed Cost (AFC) at Q = 4.

AFC = 62.5 Naira

AFC is defined as TFC / Q, so 250 / 4 = 62.5.

Calculate Average Total Cost (ATC) at Q = 4.

ATC = 89.5 Naira

ATC is the sum of AFC and AVC (62.5 + 27 = 89.5).

Key Concept

Short-run average cost relationships and cost minimization

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