\section*{Task 1 - 011221}

In international mail, for rectangular parcels (in the form of a cuboid), the following maximum and minimum dimensions are prescribed:
Maximum dimensions
Length, width, and height together \(90 \mathrm{~cm}\), with the greatest length not exceeding \(60 \mathrm{~cm}\);
Minimum dimensions Length \(10 \mathrm{~cm}\), Width \(7 \mathrm{~cm}\).

1) What is the maximum volume a parcel can have? What are the dimensions of length, width, and height in this case?

2) What is the minimum volume a parcel can have? What are the dimensions of the edges in this case? (Justification!)
}
a) If we denote the three dimensions of the cuboid with \(x, y\), and \(z\), we need to estimate the product \(x y z\) from above, given the fixed sum \(x + y + z = 90 \, \text{cm}\). For this, the inequality of the arithmetic and geometric means is particularly suitable:

\[
\sqrt[3]{x y z} \leq \frac{x + y + z}{3} ; \quad (x, y, z \geq 0)
\]

Equality holds only for \(x = y = z\),

\[
\Rightarrow V = x y z \leq \frac{(x + y + z)^{3}}{27} = 27000 \, \text{cm}^3 = 27 \, \text{l}
\]

We see that the volume can be at most 27 liters, and this is exactly the case when all dimensions are equal: \(x = y = z = 30 \, \text{cm}\) (cube).

b) This question is only a kind of "sophism." Since no specifications are made for the height, \(z\) can theoretically be \(0\), i.e., the minimum volume is zero.

\section*{Solution by Korinna Grabski}