\section*{Task 2 - 011122}

In international mail, for letter and parcel shipments in "roll form" (cylindrical shape), the following maximum and minimum dimensions are prescribed:

\begin{tabular}{ll} 
Maximum dimensions & \begin{tabular}{l} 
Length and twice the diameter together \(100 \mathrm{~cm}\), \\
Length, however, not more than \(80 \mathrm{~cm}\);
\end{tabular} \\
Minimum dimensions & \begin{tabular}{l} 
Length and twice the diameter together \(17 \mathrm{~cm}\), \\
greatest dimension not less than \(10 \mathrm{~cm}\).
\end{tabular}
\end{tabular}

1. What is the maximum volume that the shipment can have? What are the length and diameter in this case?

2. What is the minimum volume that the shipment can have? What are the length and diameter in this case?
}

a) The conditions for the maximum measure are: \(l+2 d \leq 100 \mathrm{~cm}\) and \(l \leq 80 \mathrm{~cm}\). This yields the volume of a cylinder as:

\[
V=\frac{\pi}{4} d^{2} l \leq \frac{\pi}{4} d^{2}(100 \mathrm{~cm}-2 d)=\frac{\pi}{4} \cdot 100 \mathrm{~cm} \cdot d^{2}-\frac{\pi}{2} d^{3}
\]

The necessary condition for a maximum is \(V^{\prime}(d)=0\), thus

\[
V^{\prime}(d)=\frac{\pi}{2} \cdot 100 \mathrm{~cm} \cdot d-\frac{3 \pi}{2} d^{2}=0
\]

hence \(d_{1}=0\) and \(d_{2}=\frac{100}{3} \mathrm{~cm}\). The first solution is invalid, as the volume would then be zero. The maximum length corresponding to the second solution is \(l=\frac{100}{3} \mathrm{~cm}\), with the corresponding volume \(V=29089 \mathrm{~cm}^{3}\).

It remains to show that the found solution is indeed a maximum, for which \(V^{\prime \prime}(d)<0\) is sufficient:

\[
V^{\prime \prime}(d)=\frac{\pi}{2} \cdot 100 \mathrm{~cm}-3 \pi d=-\pi \cdot 50 \mathrm{~cm}<0
\]

Thus, it is indeed a maximum. The maximum volume of the shipment is \(29089 \mathrm{~cm}^{3}\). In this case, both the diameter and the length are \(\frac{100}{3} \mathrm{~cm}\).

b) The conditions for the minimum measure do not exclude a diameter of \(0 \mathrm{~cm}\), leading to a theoretical minimum volume of \(0 \mathrm{~cm}^{3}\).

According to the first condition, the length is then at least \(17 \mathrm{~cm}\).