On vacation

Ivan and Katka discovered on vacation a regular pyramid whose base was a square with a side of 230 m and whose height was equal to the radius of a circle with the same area as the base square. Katka labelled the vertices of the square ABCD. Ivan marked on the line connecting point B with the apex of the pyramid such a point E that the length of the broken line AEC was the shortest possible. Determine the length of the broken line AEC rounded to whole centimetres.

Final Answer:

x =  416.12 m

Step-by-step explanation:

$$\begin{gathered} a=230\text{m} \\ {S}_1=a^2=230^2=52900{\text{m}}^2 \\ {S}_1=\pi r^2 \\ r=\sqrt{S_1/\pi }=\sqrt{52900/3.1416}\doteq 129.7636\text{m} \\ h=r=129.7636\doteq 129.7636\text{m} \\ u=\sqrt{2}\cdot a=\sqrt{2}\cdot 230=230\sqrt{2}\text{m}\doteq 325.2691\text{m} \\ e=\sqrt{h^2+(u/2{)}^2}=\sqrt{129.7636^2+(325.2691/2{)}^2}\doteq 208.0591\text{m} \\ {e}_2=2\cdot a=2\cdot 230=460\text{m} \\ 2\cdot e\le AEC\le e_2 \\ x=2\cdot e=2\cdot 208.0591=416.12\text{m} \end{gathered}$$

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