Question

For a manufacturing firm, the cost function is given by \(C = q^{3} + 2q^{2} + q + 1.\) The marginal and average costs at \(q = 10\) units are respectively given by:

• A. 300 and 100
• B. 340 and 125
• C. 341 and 121.1
• D. 328 and 110.1

Hint:

Calculate derivatives and divide by the quantity for average cost.

Answer 1

The Correct Option is C

The cost function is \(C = q^{3} + 2q^{2} + q + 1.\)

Marginal Cost (MC) is determined by differentiating the cost function with respect to \( q \):
\(MC = \frac{dC}{dq} = 3q^{2} + 4q + 1.\)
When \( q = 10 \), the marginal cost is calculated as:
\(MC = 3(10)^{2} + 4(10) + 1 = 300 + 40 + 1 = 341.\)

Average Cost (AC) represents the total cost divided by the quantity:
\(AC = \frac{C}{q} = \frac{q^{3} + 2q^{2} + q + 1}{q}.\)

For \( q = 10 \), the average cost is:
\(AC = \frac{10^{3} + 2(10)^{2} + 10 + 1}{10}\)

\(= \frac{1000 + 200 + 10 + 1}{10}\)

\(= \frac{1211}{10} = 121.1.\)

Therefore, the correct option is (c), \(341\) and \(121.1\).