Prism-shaped 6137

The prism-shaped vessel with a rhomboid base has one base diagonal of 10 cm and the edge of the base 14 cm.
The edge of the base and the prism height are in a ratio of 2:5.
How many liters of water is in the container when it is filled to four-fifths of the volume?

Correct answer:

V =  3.6615 l

Step-by-step explanation:

$$\begin{gathered} 1d{m}^3=1l \\ a=14/10=\frac{7}{5}=1.4\text{dm} \\ u=10/10=1\text{dm} \\ s=\frac{a+u+a}{2}=\frac{\frac{7}{5}+1+\frac{7}{5}}{2}=\frac{19}{10}=1.9\text{dm} \\ {S}_1=\sqrt{s\cdot (s-a)\cdot (s-u)\cdot (s-a)}=\sqrt{1.9\cdot (1.9-1.4)\cdot (1.9-1)\cdot (1.9-1.4)}\doteq 0.6538{\text{dm}}^2 \\ S=2\cdot S_1=2\cdot 0.6538\doteq 1.3077{\text{dm}}^2 \\ a:h=2:5 \\ h=\frac{5}{2}\cdot a=\frac{5}{2}\cdot \frac{7}{5}=\frac{5\cdot 7}{2\cdot 5}=\frac{35}{10}=\frac{7}{2}=3.5\text{dm} \\ {V}_1=h\cdot S=3.5\cdot 1.3077\doteq 4.5768\text{l} \\ V=\frac{4}{5}\cdot V_1=\frac{4}{5}\cdot 4.5768=3.6615\text{l} \end{gathered}$$

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