The funnel

The funnel has the shape of an equilateral cone. Calculate the content of the area wetted with water if you pour 3 liters of water into the funnel.

Correct result:

S =  8.7876 dm²

Solution:

$$\begin{gathered} V=3\text{ l}\to {\text{dm}}^3=3\cdot 1{\text{dm}}^3=3{\text{dm}}^3 \\ 2r=s \\ {s}^2=h^2+r^2 \\ 4r^2=h^2+r^2 \\ 3r^2=h^2 \\ h=\sqrt{3}\cdot r \\ V={\displaystyle \frac{1}{3}}\pi r^{2}h \\ V={\displaystyle \frac{1}{3}}\pi r^2\sqrt{3}\cdot r \\ V={\displaystyle \frac{\sqrt{3}}{3}}\pi r^3 \\ r=\sqrt[3]{{\displaystyle \frac{3\cdot V}{\sqrt{3}\cdot \pi }}}=\sqrt[3]{{\displaystyle \frac{3\cdot 3}{\sqrt{3}\cdot 3.1416}}}\doteq 1.1826\text{ dm } \\ h=\sqrt{3}\cdot r=\sqrt{3}\cdot 1.1826\doteq 2.0484\text{ dm } \\ s=2\cdot r=2\cdot 1.1826\doteq 2.3652\text{ dm } \\ S=\pi \cdot r\cdot s=3.1416\cdot 1.1826\cdot 2.3652=8.7876{\text{dm}}^2 \end{gathered}$$

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