617. Daily expenses for a ship's voyage consist of two parts: a constant part, equal to a rubles, and a variable part, increasing proportionally to the square of the speed. At what speed $v$ will the ship's voyage be the most economical?
Solution. Swimming will be the most economical if the costs per 1 km of the journey are the lowest. From the condition, it follows that the expenses for a day will amount to $a+k v^{3}$ ( $k$ - the proportionality coefficient); during this time, the ship will travel $24 v$ km. Therefore, the expenses per 1 km of the journey will be

$$
P=\frac{a+k v^{3}}{24 v}
$$

This function has an infinite discontinuity at $v=0$, but zero speed is not of interest to us.

Let's find the derivative: $P_{y}^{\prime}=\frac{2 k v^{3}-a}{24 v^{2}}$. The critical value of $v$ will be obtained by solving the equation $P_{v}^{\prime}=0$, i.e., $v=\sqrt[3]{\frac{a}{2 k}}$. When passing through this value, $P^{\prime}$ changes sign from minus to plus; therefore, the function has a minimum.

Thus, the most economical speed of navigation is $v=\sqrt[3]{\frac{a}{2 k}}$.