B1. Mirko performs in the circus in the "cannon launch" act. His motion, which is shown in the figure, is described by the function $f$ with the rule

$$
f(x)=x-\frac{1}{50} x^{2}
$$

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-07.jpg?height=277&width=825&top_left_y=224&top_left_x=1092)

a) How far from the launch point should the center of the net be placed so that Mirko lands safely in it?

b) What is the maximum height above the ground that Mirko reaches?
B1. a) The center of the net is at a positive zero of the function $f$, which is the solution to the equation $x-\frac{1}{50} x^{2}=0$. The solutions are $x_{1}=0$ and $x_{2}=50$. Therefore, the center of the net is 50 meters away.

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-15.jpg?height=60&width=1593&top_left_y=1935&top_left_x=320)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-15.jpg?height=57&width=1593&top_left_y=1985&top_left_x=320)

Answer: The center of the net is 50 meters away ...................................................

b) The maximum height that Mirko reaches is the maximum of the given function $f$. We calculate the second coordinate of the vertex $q=12.5$. The maximum height that Mirko reaches is $12.5 \mathrm{~m}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-15.jpg?height=68&width=1593&top_left_y=2282&top_left_x=320)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-15.jpg?height=63&width=1597&top_left_y=2333&top_left_x=318)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-15.jpg?height=60&width=1593&top_left_y=2391&top_left_x=320)
\footnotetext{
or

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-15.jpg?height=75&width=1593&top_left_y=2541&top_left_x=320)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-15.jpg?height=68&width=1593&top_left_y=2593&top_left_x=320)

![](https://cdn.mathpix.com/cropped/2024_06_07_d6b556bd14f8001ca6fag-15.jpg?height=60&width=1591&top_left_y=2648&top_left_x=324)