The variable cost (V) of manufacturing a product varies according to the…
2011
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)?
- A.
5
- B.
4
- C.
7
- D.
6
Attempted by 34 students.
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Correct answer: A
Solution:
Let the total cost be C(q) = 4q + 100/q.
Differentiate: C'(q) = 4 - 100/q^2.
Set derivative to zero to find critical points: 4 - 100/q^2 = 0 ⇒ q^2 = 25 ⇒ q = 5 (take positive root since quantity must be positive).
Verify it is a minimum: C''(q) = 200/q^3, which is positive at q = 5, so C has a minimum there.
Therefore the total cost is minimized when 5 units are produced.