The total cost (in Rs.) of producing x units of a product is \(C(x) = 0.5x^2 + 4x + 100\). The output is increasing at the rate of 3 units per day. The rate at which the total cost is increasing when the output is 10 units is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Rs. 42 per day
Both C and x change with time t. By the chain rule:
\( \displaystyle \frac{dC}{dt} = \frac{dC}{dx} \times \frac{dx}{dt} \)
\( \displaystyle \frac{dC}{dx} = x + 4 \)
At x = 10, with \(\frac{dx}{dt} = 3\):
\( \displaystyle \frac{dC}{dt} = (10 + 4) \times 3 = 42 \)
So the total cost is increasing at Rs. 42 per day.
Why the other options are incorrect
Option (1) Rs. 14 per day — Incorrect. This is only the marginal cost \(\frac{dC}{dx}\) at x = 10 (Rs. 14 per unit); it is not yet multiplied by the 3 units per day.
Option (3) Rs. 30 per day — Incorrect. The true rate is (10 + 4) × 3 = 42.
Option (4) Rs. 190 per day — Incorrect. This is the total cost C(10) = 50 + 40 + 100, not its rate of change.