GATE CS 2011 Q.63 (GA Q.63): Quantitative Aptitude

The variable cost ( $V$ ) of manufacturing a product varies according to the equation $V = 4q$ , where $q$ is the quantity produced. The fixed cost ( $F$ ) of production of same product reduces with $q$ according to the equation $F = 100/q$ . How many units should be produced to minimize the total cost ( $V + F$ )?

Official answer and solution

Official answer: A
From: Official GATE 2011 answer key

Solution

Formulate total cost as a function of quantity q and find the critical point that minimizes it.

The total manufacturing cost $C(q)$ is the sum of variable cost $V$ and fixed cost $F$ :

C(q) = V + F = 4q + \frac{100}{q} \quad (q > 0)

To find the quantity $q$ that minimizes total cost, differentiate $C(q)$ with respect to $q$ and set the derivative to zero:

\frac{dC}{dq} = 4 - \frac{100}{q^2} = 0

4q^2 = 100 \implies q^2 = 25 \implies q = 5

Check the second derivative for convexity:

\frac{d^2C}{dq^2} = \frac{200}{q^3}

At $q = 5$ , $\frac{d^2C}{dq^2} = \frac{200}{125} = 1.6 > 0$ , confirming that $q = 5$ achieves the minimum total cost.

Thus, exactly 5 units should be produced.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2011 Computer Science and Information Technology (CS)
Question: 63 of 65, printed as GA Q.63
Section: General Aptitude

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