## Task 1 - 110921

For a right circular cylinder, the measures of the circumference of its base (in $\mathrm{cm}$), the area of its lateral surface (in $\mathrm{cm}^{2}$), and its volume (in $\mathrm{cm}^{3}$) are to be equal to each other. Determine the radius of the base circle and the height of each such cylinder!
Assuming a right circular cylinder meets the conditions of the problem. Let $r$ be the measure of the radius of its base (in centimeters) and $h$ the measure of its height.

Then the measure of the circumference of its base (in $\mathrm{cm}$): $2 \pi r$, the measure of the area of its lateral surface: $2 \pi r \cdot h$, and the measure of its volume (in $\mathrm{cm}^{3}$): $\pi r^{2} \cdot h$, and it holds that: $2 \pi r=2 \pi r \cdot h$, from which, due to $r>0$, $h=1$ follows.

Furthermore, it holds that: $2 \pi r \cdot h=\pi r^{2} \cdot h$, from which, due to $r=2$, follows.

Therefore, at most a right circular cylinder with a radius of $2 \mathrm{~cm}$ and a height of 1 cm can meet the conditions of the problem.

In this case, the circumference of the base is $4 \pi \mathrm{cm}$, the lateral surface area is $4 \pi \mathrm{cm}^{2}$, and the volume is $4 \pi \mathrm{cm}^{3}$.

## Solution adopted from $[5]$