The fuel cost per hour for running a ship is proportional to the square of the speed generated in…

Since the fuel cost is proportional to the square of the speed: $C_f = kv^2$. Given that $C_f = 75$ when $v = 10$: $75 = k(10)^2$, $k = 0.75$.

Total cost $= 0.75v^2 + 1000$ Cost per unit distance $= C_d = \frac{0.75v^2 + 1000}{v} = 0.75v + \frac{1000}{v}$

For most economical speed cost per unit distance should be minimum. $C_d' = 0.75 - \frac{1000}{v^2}$, $C_d' = 0 \Rightarrow v^2 = \frac{1000}{0.75}$, $v = 36.51$; $C_d'' = \frac{2000}{v^3} > 0$, $v = 36.51$ is point of minima. Most economical speed is 36.51 knots.