Octagonal prism vase

A total of 0.7 litres of water can be poured into an octagonal prism vase. The vase has a base area of 25 cm² and a wall thickness of 12 mm. What is the height of the vase?

Final Answer:

h =  88.301 cm

Step-by-step explanation:

$$\begin{gathered} V=0.7\text{l}\to {\text{cm}}^3=0.7\cdot 1000{\text{cm}}^3=700{\text{cm}}^3 \\ S=25{\text{cm}}^2 \\ s=12\text{mm}\to \text{cm}=12:10\text{cm}=1.2\text{cm} \\ n=8 \\ S=2(1+\sqrt{2})\cdot a^2 \\ a=\sqrt{\frac{S}{2\cdot (1+\sqrt{2})}}=\sqrt{\frac{25}{2\cdot (1+\sqrt{2})}}\doteq 2.2754\text{cm} \\ S/n=\frac{a\cdot h_1}{2} \\ {h}_1=\frac{2\cdot S}{n\cdot a}=\frac{2\cdot 25}{8\cdot 2.2754}\doteq 2.7467\text{cm} \\ {h}_2=h_1-s=2.7467-1.2\doteq 1.5467\text{cm} \\ a:h_1=b:h_2 \\ b=a\cdot \frac{h_2}{h_1}=2.2754\cdot \frac{1.5467}{2.7467}\doteq 1.2813\text{cm} \\ {S}_2=n\cdot \frac{b\cdot h_2}{2}=8\cdot \frac{1.2813\cdot 1.5467}{2}\doteq 7.9274{\text{cm}}^2 \\ V=S_{2}h \\ h=V/S_2=700/7.9274=88.301\text{cm} \end{gathered}$$

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