3. Janez and Peter, who are $450 \mathrm{~m}$ apart from each other, start walking towards each other at the same time. Janez has his eyes closed and moves slowly - in the first minute, he walks $5 \mathrm{~m}$, and in each subsequent minute, he walks $15 \mathrm{~m}$ more than in the previous minute. Peter walks $100 \mathrm{~m}$ in the first minute, and in each subsequent minute, he walks $10 \mathrm{~m}$ less than in the previous minute. After how much time will they meet? Write down your answer.
3. Let's create a table in which we record the distance traveled in meters.

|  | Janez | Peter | B o t h t o t a l |  |
| :--- | :---: | :---: | :---: | :---: |
| in the last minute |  |  |  |  |
| 1st minute | 5 | 100 | 105 | 105 |
| 2nd minute | 20 | 90 | 110 | 215 |
| 3rd minute | 35 | 80 | 115 | 330 |
| 4th minute | 50 | 70 | 120 | 450 |

Janez and Peter will meet after 4 minutes.

We can also solve the problem in another way. The distances traveled by Janez in consecutive minutes form the terms of an arithmetic sequence with \(a_{1}=5\) and \(d=15\). Similarly, for Peter's distance: \(a_{1}=100\), \(d=-10\). The total distance traveled by Janez in \(t\) minutes is \(\frac{t}{2}(10+15(t-1))\), and the total distance traveled by Peter is \(\frac{t}{2}(200-(t-1) \cdot 10)\). Since they together travel \(450 \mathrm{~m}\), we write the equation \(450=\frac{t}{2}(10+15(t-1))+\frac{t}{2}(200-(t-1) \cdot 10)\), which simplifies to \(t^{2}+41 t-180=0\). The positive solution to this equation is \(t=4\), meaning they will meet after 4 minutes.

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-12.jpg?height=60&width=1642&top_left_y=2363&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-12.jpg?height=51&width=1639&top_left_y=2419&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-12.jpg?height=102&width=1640&top_left_y=2459&top_left_x=205)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-12.jpg?height=60&width=1636&top_left_y=2540&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-12.jpg?height=48&width=1639&top_left_y=2603&top_left_x=206)

Answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 point