3. Gavriil saw a squirrel sitting on a tree branch right in front of him at a distance of 3 m 75 cm through the window. He decided to feed the little animal and threw a nut horizontally at a speed of 5 m/s directly towards the squirrel. Will the squirrel be able to catch the nut if it can jump in any direction at a speed over a distance of 1 m 70 cm? Assume the acceleration due to gravity is $g=10 \mathrm{~m} / \mathrm{c}^{2}$, and neglect air resistance.
Answer: No. Solution: The coordinates of the squirrel at the initial moment: $x=a ; y=0$. The nut moves according to the law: $x(t)=V_{0} t ; y(t)=\frac{g t^{2}}{2}$. Therefore, the square of the distance from the sitting squirrel to the flying nut at time $t$ is: $r^{2}=\left(V_{0} t-a\right)^{2}+\left(\frac{g t^{2}}{2}\right)^{2}=\frac{g^{2} t^{4}}{4}+V_{0}^{2} t^{2}-2 V_{0} a t+a^{2}$. Substituting numerical values, we get: $r^{2}=25 t^{4}+25 t^{2}-\frac{75}{2} t+\frac{225}{16}=\frac{25}{16}\left(16 t^{4}+16 t^{2}-24 t+9\right)$.

We need to minimize the function $f(t)=16 t^{4}+16 t^{2}-24 t+9$. Since $f^{\prime}(t)=64 t^{3}+32 t-24=8\left(8 t^{3}+4 t-3\right)$, we solve the equation $8 t^{3}+4 t-3=0$. Since $8 t^{3}+4 t-3$ $=(2 t-1)\left(4 t^{2}+2 t+3\right)$ and the equation $4 t^{2}+2 t+3=0$ has no real roots, the equation $8 t^{3}+4 t-3=0$ has a unique root $t=\frac{1}{2}$. This will be the minimum point of the function $f(t)=16 t^{4}+16 t^{2}-24 t+9$, as the derivative changes sign from negative to positive at this point.

Therefore, $\left(r^{2}\right)_{\min }=\frac{25}{16}\left(\frac{16}{16}+\frac{16}{4}-\frac{24}{2}+9\right)=\frac{25 \cdot 2}{16}$, and $r_{\min }=\frac{5 \sqrt{2}}{4}(\mathrm{m})$.

We need to compare this number with $1.7=\frac{17}{10}$. Since $\frac{5 \sqrt{2}}{4}>\frac{17}{10} \Leftrightarrow 25 \sqrt{2}>34 \Leftrightarrow 25>17 \sqrt{2} \Leftrightarrow$ $625>289 \cdot 2$, the nut will be out of reach for the squirrel.

Grading criteria: 20 points: correct solution, correct value of the minimum distance, and correct answer to the question; 15 points: with correct solution and correct answer, there are defects; for example: the root of the cubic equation is found, but it is not proven that there are no other roots; the zero of the derivative is found, but it is not proven that it is a minimum point; the comparison of numbers at the end is conducted incorrectly, etc.; 10 points: correct approach to solving the problem, but the answer is incorrect due to computational errors; 5 points: correct start of the solution, the fourth-degree equation is obtained and the derivative (by $t$ or by $x$) is correctly obtained, but its zeros are not found or are found incorrectly.