The cost of running a bus from A to B is Rs. $$(av + b/v)$$ where v km/h is the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be Rs. 75 while at 40 km/h, it is Rs. 65. Then the most economical speed (in km/h) of the bus is :

The cost of running the bus is given by the expression $$ av + \frac{b}{v} $$, where $$ v $$ is the speed in km/h. We are provided with two conditions:

When $$ v = 30 $$ km/h, cost = Rs. 75. This gives the equation:

$$ 30a + \frac{b}{30} = 75 \quad \text{(Equation 1)} $$

When $$ v = 40 $$ km/h, cost = Rs. 65. This gives the equation:

$$ 40a + \frac{b}{40} = 65 \quad \text{(Equation 2)} $$

To eliminate denominators, multiply Equation 1 by 30:

$$ 30 \times 30a + 30 \times \frac{b}{30} = 30 \times 75 $$

$$ 900a + b = 2250 \quad \text{(Equation 3)} $$

Multiply Equation 2 by 40:

$$ 40 \times 40a + 40 \times \frac{b}{40} = 40 \times 65 $$

$$ 1600a + b = 2600 \quad \text{(Equation 4)} $$

Subtract Equation 3 from Equation 4:

$$ (1600a + b) - (900a + b) = 2600 - 2250 $$

$$ 1600a + b - 900a - b = 350 $$

$$ 700a = 350 $$

$$ a = \frac{350}{700} = \frac{1}{2} = 0.5 $$

Substitute $$ a = 0.5 $$ into Equation 3:

$$ 900 \times 0.5 + b = 2250 $$

$$ 450 + b = 2250 $$

$$ b = 2250 - 450 = 1800 $$

The cost function is $$ C(v) = 0.5v + \frac{1800}{v} $$. To find the speed that minimizes cost, take the derivative and set it to zero.

First derivative:

$$ C'(v) = \frac{d}{dv} \left( 0.5v + 1800v^{-1} \right) = 0.5 - \frac{1800}{v^2} $$

Set $$ C'(v) = 0 $$:

$$ 0.5 - \frac{1800}{v^2} = 0 $$

$$ 0.5 = \frac{1800}{v^2} $$

$$ v^2 = \frac{1800}{0.5} = 1800 \times 2 = 3600 $$

$$ v = \sqrt{3600} = 60 \quad \text{(since speed must be positive)} $$

Verify it is a minimum using the second derivative test:

$$ C''(v) = \frac{d}{dv} \left( 0.5 - 1800v^{-2} \right) = 0 + 3600v^{-3} = \frac{3600}{v^3} $$

For $$ v = 60 > 0 $$, $$ C''(60) = \frac{3600}{60^3} > 0 $$, confirming a minimum.

Thus, the most economical speed is 60 km/h. Comparing with the options:

A. 45

B. 50

C. 60

D. 40

Hence, the correct answer is Option C.