6. Try to advance as far as possible in the analytical solution of the problem given below. In case $B$ is necessary, a computer may be used at the final stage.

Point $A$ is located on a meadow, point $B$ - on a sandy wasteland. The distance between the points is 24 km. The boundary between the wasteland and the meadow is a straight line. The distance from point $A$ to the boundary is 8 km, and the distance from point $B$ to the boundary is 4 km. Find the minimum time it takes for a pedestrian to get from point $A$ to point $B$, if their maximum speed on the wasteland is 3 km/hour, and on the meadow 6 km/hour.
Solution. Our task: find such a point $C$ on $A^{\prime} B^{\prime}$ (see the figure) so that the path along the trajectory $A C + C B$ takes the minimum possible time. The distance $A^{\prime} B^{\prime}$ is $12 \sqrt{3}$, let $A^{\prime} C = x$, where $x \in [0 ; 12 \sqrt{3}]$ (in fact, it is obvious that $x \in [8 \sqrt{3} ; 12 \sqrt{3}]$).

Then the travel time $t$ is

![](https://cdn.mathpix.com/cropped/2024_05_06_ff400db63ff6b9f02cbeg-4.jpg?height=403&width=606&top_left_y=198&top_left_x=1296)

$$
\begin{gathered}
\frac{A C}{V_{1}} + \frac{C B}{V_{2}} = \frac{\sqrt{x^{2} + 64}}{6} + \frac{\sqrt{(12 \sqrt{3} - x)^{2} + 16}}{3}. \text{ Thus, we need to find the minimum of the function } \\
f(x) = \frac{1}{6} \left( \sqrt{x^{2} + 64} + 2 \sqrt{(12 \sqrt{3} - x)^{2} + 16} \right).
\end{gathered}
$$

The first approach involves using the derivative. The derivative

$f^{\prime}(x) = \frac{1}{6} \left( \frac{x}{\sqrt{x^{2} + 64}} - \frac{2(12 \sqrt{3} - x)}{\sqrt{(12 \sqrt{3} - x)^{2} + 16}} \right)$ is zero when $\frac{x}{\sqrt{x^{2} + 64}} = \frac{2(12 \sqrt{3} - x)}{\sqrt{(12 \sqrt{3} - x)^{2} + 16}}$.

We need to find the roots of this equation.

The second approach is based on the analogy with the propagation of light. The ratio

$\frac{V_{1}}{\sin \alpha} = \frac{V_{2}}{\sin \beta}$ in our notation is written as $\frac{6 \sqrt{x^{2} + 64}}{x} = \frac{3 \sqrt{(12 \sqrt{3} - x)^{2} + 16}}{12 \sqrt{3} - x}$, which is equivalent to (2).

Equation (2) can be written as $x^{2} \left( (12 \sqrt{3} - x)^{2} + 16 \right) = 4 (12 \sqrt{3} - x)^{2} \left( x^{2} + 64 \right)$

$\Leftrightarrow x^{4} - 24 \sqrt{3} x^{3} + 512 x^{2} - 2048 \sqrt{3} x + 36864 = 0$.

This algebraic equation cannot be solved analytically. At this point, a computer should be used. The approximate roots are $x_{1} = 18.7138$ and $x_{2} = 22.9267$ (not suitable, as this value is greater than $12 \sqrt{3}$). This numerical solution (sufficient accuracy $x = 18.7$) can be obtained by participants using a calculator or any computer programs: both packaged (Excel, Mathematica, WolframAlpha, etc.) and those written independently. A detailed description of the process is not required.

Substituting $x = 18.7138$ into (1) gives the time $t = 4.89$ hours (or 4 hours 53 minutes 24 seconds).

Answer: 4.89 hours (4.9 hours is acceptable) or 4 hours 53 minutes (4 hours 54 minutes is acceptable).

Criteria: 20 points - correct (not necessarily the same as above) solution and correct answer; 15 points - mostly correct solution, but the answer differs from the correct one by more than 30 seconds; 10 points - correct answer, but a computer is used at an early stage of the solution (for example, using it to find the minimum of function (1) or the zeros of function (2)) - thereby violating the condition: "try to advance as far as possible in the analytical solution of the problem"; 5 points - correct approach to the solution, but the answer is either missing or incorrect; $\mathbf{0}$ points - everything else.

Attention! The final score of the participant is the sum of the points for five tasks out of six, that is, the worst of the six grades for the tasks is not included in the total score.