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 the derivative of the cost function with respect to \( q \):
 \(MC = \frac{dC}{dq} = 3q^{2} + 4q + 1.\)
Substitute \( q = 10 \): 
\(MC = 3(10)^{2} + 4(10) + 1 = 300 + 40 + 1 = 341.\)

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

Substitute \( q = 10 \): 
\( AC = \frac{10^{3} + 2(10)^{2} + 10 + 1}{10} \)

= \(\frac{1000 + 200 + 10 + 1}{10} \)

\(= \frac{1211}{10} \)
\(= 121.1.\)

Hence, the correct answer is (c) \( 341 \) and \( 121.1 \).