A car can be rented under the following conditions. Base fee: 50 fillers per km. Speed surcharge: above 30 km/h, 3 fillers per km/h per km. Time surcharge: beyond 10 hours, 1 Ft per minute. For a 500 km trip, at what average speed should one travel to minimize the car rental cost?
Let the sought speed be $x \mathrm{~km} /$ hour. (Since the distance is greater than $300 \mathrm{~km}$, therefore $x \geqq 30$.)

The total rent to be paid in this case is

$$
\begin{aligned}
500 \cdot 0.5 + 500(x-30) \cdot 0.03 + & \left(\frac{500}{x}-10\right) 60 = 250 + 15x - 450 + \frac{30000}{x} - 600 = \\
= & 15x + \frac{30000}{x} - 800
\end{aligned}
$$

Thus, we are looking for the minimum of the function

$$
y=15x + \frac{30000}{x}
$$

Our function is a sum of two terms, the product of which is constant. It is known that the sum of two numbers - provided their product is constant - is minimal when the two numbers are equal, i.e.,

$$
15x = \frac{30000}{x}
$$

which means

$$
x^2 = 2000
$$

from which

$$
x = \sqrt{2000} \approx 44.7 \mathrm{~km} / \mathrm{h}
$$

Jenő Plichta (Mezőkövesd, I. László g. 1V. o. t.)