## Task 2 - 221222

Investigate whether among all triangles for which the side lengths $a, b, c$ satisfy the relationships $a \leq 1 \mathrm{~cm} \leq b \leq 2 \mathrm{~cm} \leq c \leq 3 \mathrm{~cm}$, there is a triangle with the maximum possible area.

If this is the case, determine this area.
According to the trigonometric area formula

$$
A=\frac{1}{2} a b \sin \gamma
$$

for a triangle with side lengths $a$ and $b$, and the included angle $\gamma$, $A$ obviously takes its maximum under the consideration of the constraints

$$
a \in(0,1], b \in[1,2], \gamma \in(0, \pi)
$$

for

$$
a=1, b=2, \gamma=\frac{\pi}{2}
$$

However, we have an additional constraint here, namely $c \in[2,3]$ for the third side $c$, which is also satisfied because

$$
c=\sqrt{1^{2}+2^{2}}=\sqrt{5} \approx 2.236
$$

Thus, this right-angled triangle with an area of $A=1$ indeed represents the solution to the problem.