## Task 4 - 240824

A metal sheet has the shape shown in the figure, where $a, b$, and $x$ are given lengths. The sheet is to be divided along the dashed line into two parts, and from each part, a cuboid box of height $x$ is to be made, open at the top.

1. Calculate the volume of such a

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0750.jpg?height=286&width=817&top_left_y=1356&top_left_x=311)
box if $a=360 \mathrm{~mm}, b=120 \mathrm{~mm}$, $x=25 \mathrm{~mm}$ are given!

2. Determine the volume of such a box, expressed in terms of the variables $a, b$, and $x$, which can only take positive values due to their meaning as lengths!
3. Let arbitrary positive values $a$ and $b$ be given.

Determine, in dependence on these $a, b$, all the values of the variable $x$ that make it possible to construct boxes of the described type!
1. Each of the two parts has a shape that results from a rectangle with side lengths $\frac{a}{2}=180 \mathrm{~mm}, \mathrm{~b}=$ $120 \mathrm{~mm}$ by cutting out squares with side length $x=25 \mathrm{~mm}$ at all four corners.

A cuboid-shaped open-top box of height $x=25 \mathrm{~mm}$, made from this, has a base that is a rectangle with side lengths $\frac{a}{2}-2 x=130 \mathrm{~mm}, b-2 x=70 \mathrm{~mm}$. Therefore, its volume is

$$
V=130 \mathrm{~mm} \cdot 70 \mathrm{~mm} \cdot 25 \mathrm{~mm}=227500 \mathrm{~mm}^{3}
$$

2. By the same reasoning as in 1., we get $V=\left(\frac{a}{2}-2 x\right) \cdot(b-2 x) \cdot x$.
3. As a length, $x$ must be positive. Furthermore, the construction of the mentioned boxes is only possible if $\frac{a}{2}-2 x$ and $b-2 x$ are also positive as length measurements (for the edge lengths of a box), i.e., if in addition to the inequality $x>0$, the inequalities

$$
\frac{a}{2}-2 x>0 \quad \text { and } \quad b-2 x>0
$$

hold. These are equivalent to $x<\frac{a}{4}$ and $x<\frac{b}{2}$

The sought values are thus all positive values of $x$ that are smaller than the smaller of the two length measurements $\frac{a}{4}, \frac{b}{2}$.

Solutions of the II. Round 1984 adopted from [5]

### 5.26.3 III. Round 1984, Class 8