## 3 The Case Study
#### Mathematical Modeling--Relations  
In this case study, the two bottles are assumed to have the same volume. Thus,  
\[\mathbf{v}_{1}=\mathbf{v}_{2} \tag{1}\]  
The volume for the initial bottle is the following (Figure 1):  
\[v_{1}=\pi*{p_{1a}}^{2}*h_{1a}+\pi*p^{2}*h+\frac{1}{3}*\pi*p^{2}*h_{2a}+\frac{2 }{3}*\pi*{p_{1a}}^{2}*h_{2a} \tag{2}\]  
While the volume of the final bottle is the following (Figure 2):  
\[v_{2}=\pi*{p_{k}}^{2}*h_{k}+\pi*p^{2}*h \tag{3}\]For the next step, the area for both bottles should be estimated in order to demonstrate the advantages or disadvantages of redesigning in terms of material usage. Thus, the area for the initial bottle is the following:  
\[e_{1}=\left( \begin{array}{c}2*\pi*p*h+\pi*p^{2}+2*\pi*p_{1a}*h_{1a}+\\ +\pi*p_{1a}^{2}+\pi*p_{1a}*h_{2a}*\\ \left(\begin{array}{c}\sqrt{\frac{{{h_{2a}}^{2}}*{p^{2}}+{{p^{2}}}-{{p^{2}}} {{p^{2}}}}}+\\ +\left(\frac{{{h_{2a}}*{p_{1a}}*{Arcsin}\left(\frac{{{\sqrt{{h_{2a}}^{2}}+{{p^{2} }}-{{p^{2}}}{{p^{2}}}}}\sqrt{{-{p^{2}}}+{{p_{1a}}^{2}}}}}{{{2{p^{2}}}+{{p^{2}}}- {{p^{2}}}{{p^{2}}}{{p^{2}}}}-{{p^{2}}+{{p_{1a}}^{2}}}}}}\\ \sqrt{{{h_{2a}}^{2}}+{{p^{2}}}-{{p_{1a}}^{2}}}\sqrt{-{{p^{2}}}+{{p_{1a}}^{2}}} }\end{array}\right)\end{array} \right)\right)10^{-6} \tag{4}\]  
And the area for the final bottle is the following:  
\[e_{2}=\left(2*\pi*p_{k}*h_{k}+2*\pi*{{p_{k}}^{2}}+2*\pi*p*h\right)10^{-6} \tag{5}\]  
In order to calculate the dimensions of the final bottle, while maintaining the same volume in both bottles, the radius of the final bottle is derived by the following relation. \[p_{k}=\frac{\sqrt{{{h_{2a}}*{p^{2}}+3*{{h_{1a}}*{{p_{1a}}^{2}}}+2*{{h_{2a}}*{{ p_{1a}}^{2}}}}}}{\sqrt{3*{{h_{k}}}}} \tag{6}\]  
where  
\(h_{k}=\) total height (final bottle)  
\(h=\) neck height  
The neck for both bottles has been designed to have the same dimensions. Modifying the neck would require costly adjustments to machinery and closures, potentially disrupting operations. Keeping it unchanged allows the redesign to focus on the bottle body, enabling material and logistical improvements without affecting functionality or manufacturing efficiency. A code in Mathematica has been developed in order to calculate the relation between the height and the radius of the final bottle. The outcomes are presented in Table 1.  
As shown in Table 1, the solution in which the reduction in material usage is maximized corresponds to the final bottle with a total height of 89.9571 mm and a radius of 44.9785 mm. In this solution, the surface area of the initial bottle is estimated at 0.0428 m\({}^{2}\), while the surface area of the final bottle is 0.0393 m\({}^{2}\). The final bottle uses 8.01% less material while maintaining the same volume as the initial bottle; this reduction was estimated by comparing the surface areas of the two designs, assuming uniform wall thickness and identical material properties. On the other hand, the solution in which the height of the final bottle is 168.16 mm and its radius is 32.8975 mm results in the same surface area as the initial bottle, i.e., 0.0428 m\({}^{2}\), with zero gain in material usage. Thus, in the present case, the following cases could be defined: \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Total Height (mm)**} & \multicolumn{2}{c}{**Radius (mm)**} & \multicolumn{2}{c}{**Bottle Shoulder (mm)**} & \multirow{2}{*}{**Neck**} & \multicolumn{2}{c}{**Area (m\({}^{2}\))**} \\ \hline
**Initial** & **Final** & **Initial** & **Final** & **Initial** & **Final** & **(mm)** & **Initial** & **Final** \\
**Bottle** & **Bottle** & **Bottle** & **Bottle** & **Bottle** & **Bottle** & **Bottle** & **Bottle** & **Bottle** \\ \hline  
\hline \hline \end{tabular} \({}^{1}\) Optimum solution—Comparison between the two configurations. The production of the final bottle requires 8.011% less material. \({}^{2}\) Zero-gain solution—Both configurations have the same volume (0.00057706 m\({}^{3}\)) and the same surface area (0.0427525 m\({}^{2}\)).  
\end{table} 
Table 1: Mathematical relations.