Question

Rate of fuel consumption \( f_c \) (litres per hour) of a truck varies with truck speed \( x \), (kmph) as given below \[ f_c = 20 + \frac{x^2}{50} \] The fuel price is Rs. 70 per litre. Other costs amount to Rs. 500 per hour. If the truck travels 100 km from a coal mine to a thermal plant, the speed of the truck, in kmph, that minimizes the total cost is \(\underline{\hspace{2cm}}\) (round off to one decimal place).

Hint:

To minimize the total cost, differentiate the total cost function with respect to speed and solve for the critical points.

Answer 1

Correct Answer: 35

The total cost consists of two parts: the fuel cost and the other fixed cost. The fuel cost per hour is: \[ \text{Fuel cost} = 70 \times f_c = 70 \times \left(20 + \frac{x^2}{50}\right) \] Thus, the total cost is: \[ \text{Total cost} = \text{Fuel cost} + 500 = 70 \times \left(20 + \frac{x^2}{50}\right) + 500 \] To minimize the total cost, we first calculate the total time to travel 100 km, which is: \[ \text{Time} = \frac{100}{x} \] Thus, the total cost as a function of \( x \) becomes: \[ \text{Total cost}(x) = \left(70 \times \left(20 + \frac{x^2}{50}\right) + 500 \right) \times \frac{100}{x} \] Now, minimize this function by taking its derivative and setting it equal to zero. After differentiation and solving, we get the optimal speed as: \[ \boxed{35.0} \, \text{kmph} \]