The problem asks for the output level (Q) where Average Cost (AC) equals Marginal Cost (MC), given the Total Cost (TC) function: $TC = 200 + 5Q + 2Q^2$. This is a key concept in microeconomics related to a firm's cost structure.

Cost Function Analysis

We are given the Total Cost (TC) function:

$TC = 200 + 5Q + 2Q^2$

Calculating Average Cost (AC)

Average Cost (AC) is calculated by dividing the Total Cost (TC) by the quantity of output (Q).

$AC = \frac{TC}{Q} = \frac{200 + 5Q + 2Q^2}{Q}$

Simplifying this gives:

$AC = \frac{200}{Q} + 5 + 2Q$

Calculating Marginal Cost (MC)

Marginal Cost (MC) is the change in Total Cost (TC) resulting from producing one additional unit of output. It is the derivative of the TC function with respect to Q.

$MC = \frac{d(TC)}{dQ} = \frac{d(200 + 5Q + 2Q^2)}{dQ}$

Calculating the derivative:

$MC = 0 + 5 + 2(2Q) = 5 + 4Q$

Finding the Output Level where AC = MC

To find the output level where AC equals MC, we set the two expressions equal to each other:

$AC = MC$

$\frac{200}{Q} + 5 + 2Q = 5 + 4Q$

Solving for Q

Now, we solve the equation for Q:

• Subtract 5 from both sides: $\frac{200}{Q} + 2Q = 4Q$
• Subtract 2Q from both sides: $\frac{200}{Q} = 2Q$
• Multiply both sides by Q (assuming $Q \neq 0$): $200 = 2Q^2$
• Divide both sides by 2: $100 = Q^2$
• Take the square root of both sides: $Q = \sqrt{100}$
• Since output (Q) must be positive: $Q = 10$

Therefore, the output level at which Average Cost (AC) and Marginal Cost (MC) are equal is 10 units.