36 км - это 1 - 2/5 пути = 5/5 - 2/5 = 3/5
тогда, 36 / 3 * 5 - весь путь
ответ 60 км.
1000*10=10000см²
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может я ошибаюсь?
∑ - знак суммы, обычно применяется в математике.
![\cos \pi x\cdot \sqrt{1-ctg^22\pi x} +\sin \pi x= \sqrt{2} \\ \\ \cos \pi x\cdot \sqrt{1-(ctg^22\pi x+1-1)} +\sin \pi x= \sqrt{2} \\ \\ \cos \pi x\cdot \sqrt{2- \frac{1}{\sin^22\pi x} } +\sin \pi x= \sqrt{2} \\ \\ \sqrt{2\cos^2\pi x- \frac{1}{4\sin^2\pi x} } +\sin\pi x= \sqrt{2} \\ \\ \sqrt{2-2\sin^2\pi x- \frac{1}{4\sin^2\pi x} } +\sin\pi x= \sqrt{2}](https://tex.z-dn.net/?f=%5Ccos++%5Cpi+x%5Ccdot+%5Csqrt%7B1-ctg%5E22%5Cpi+x%7D+%2B%5Csin+%5Cpi+x%3D+%5Csqrt%7B2%7D+%5C%5C+%5C%5C+%5Ccos+%5Cpi+x%5Ccdot+%5Csqrt%7B1-%28ctg%5E22%5Cpi+x%2B1-1%29%7D+%2B%5Csin+%5Cpi+x%3D+%5Csqrt%7B2%7D+%5C%5C+%5C%5C+%5Ccos+%5Cpi+x%5Ccdot+%5Csqrt%7B2-+%5Cfrac%7B1%7D%7B%5Csin%5E22%5Cpi+x%7D+%7D+%2B%5Csin+%5Cpi+x%3D+%5Csqrt%7B2%7D+%5C%5C+%5C%5C++%5Csqrt%7B2%5Ccos%5E2%5Cpi+x-+%5Cfrac%7B1%7D%7B4%5Csin%5E2%5Cpi+x%7D+%7D+%2B%5Csin%5Cpi+x%3D+%5Csqrt%7B2%7D+%5C%5C+%5C%5C++%5Csqrt%7B2-2%5Csin%5E2%5Cpi+x-+%5Cfrac%7B1%7D%7B4%5Csin%5E2%5Cpi+x%7D+%7D+%2B%5Csin%5Cpi+x%3D+%5Csqrt%7B2%7D+)
Сделаем замену.
Пусть
![\sin \pi x=t](https://tex.z-dn.net/?f=%5Csin+%5Cpi+x%3Dt)
, причем
![|t| \leq 1](https://tex.z-dn.net/?f=%7Ct%7C+%5Cleq+1)
. Тогда
![\sqrt{2-2t^2- \frac{1}{4t^2} } +t= \sqrt{2} \\ \\ \sqrt{ \frac{8t^2-8t^4-1}{4t^2} } = \sqrt{2} -t](https://tex.z-dn.net/?f=+%5Csqrt%7B2-2t%5E2-+%5Cfrac%7B1%7D%7B4t%5E2%7D+%7D+%2Bt%3D+%5Csqrt%7B2%7D+%5C%5C+%5C%5C++%5Csqrt%7B+%5Cfrac%7B8t%5E2-8t%5E4-1%7D%7B4t%5E2%7D+%7D+%3D+%5Csqrt%7B2%7D+-t)
Возведем обе части в квадрат, получаем:
![\frac{8t^2-8t^4-1}{4t^2} =2+t^2-2t \sqrt{2} |\cdot 4t^2\\ \\ 8t^2-8t^4-1=8t^2+4t^4-8 t\sqrt{2} \\ \\ 12t^4-8t\sqrt{2} +1=0](https://tex.z-dn.net/?f=+%5Cfrac%7B8t%5E2-8t%5E4-1%7D%7B4t%5E2%7D+%3D2%2Bt%5E2-2t+%5Csqrt%7B2%7D+%7C%5Ccdot+4t%5E2%5C%5C+%5C%5C+8t%5E2-8t%5E4-1%3D8t%5E2%2B4t%5E4-8+t%5Csqrt%7B2%7D+%5C%5C+%5C%5C+12t%5E4-8t%5Csqrt%7B2%7D+%2B1%3D0)
Положим
![z=\sqrt{2} t](https://tex.z-dn.net/?f=z%3D%5Csqrt%7B2%7D+t)
. Тогда
![3z^4-4z^3+1=0\\ 3z^4-3z^3-z^3+z^2-z^2+z-z+1=0\\ 3z^3(z-1)-z^2(z-1)-z(z-1)-(z-1)=0\\ (z-1)(3z^3-z^2-z-1)=0\\ z-1=0;\,\,\,\Rightarrow \,\,\,z=1;\\ \\ 3z^3-z^2-z-1=0](https://tex.z-dn.net/?f=3z%5E4-4z%5E3%2B1%3D0%5C%5C+3z%5E4-3z%5E3-z%5E3%2Bz%5E2-z%5E2%2Bz-z%2B1%3D0%5C%5C+3z%5E3%28z-1%29-z%5E2%28z-1%29-z%28z-1%29-%28z-1%29%3D0%5C%5C+%28z-1%29%283z%5E3-z%5E2-z-1%29%3D0%5C%5C+z-1%3D0%3B%5C%2C%5C%2C%5C%2C%5CRightarrow+%5C%2C%5C%2C%5C%2Cz%3D1%3B%5C%5C+%5C%5C+3z%5E3-z%5E2-z-1%3D0)
Добавим и вычтем некоторые слагаемые
![3z^3-3z^2+2z^2-2z+z-1=0\\ 3z^2(z-1)+2z(z-1)+z-1=0\\ (z-1)(3z^2+2z+1)=0\\z-1=0;\,\,\,\,\Rightarrow\,\,\,z=1\\ \\ 3z^2+2z+1=0\\D=b^2-4ac=2^2-4\cdot3\cdot1=4-12=-8\ \textless \ 0](https://tex.z-dn.net/?f=3z%5E3-3z%5E2%2B2z%5E2-2z%2Bz-1%3D0%5C%5C+3z%5E2%28z-1%29%2B2z%28z-1%29%2Bz-1%3D0%5C%5C+%28z-1%29%283z%5E2%2B2z%2B1%29%3D0%5C%5Cz-1%3D0%3B%5C%2C%5C%2C%5C%2C%5C%2C%5CRightarrow%5C%2C%5C%2C%5C%2Cz%3D1%5C%5C+%5C%5C+3z%5E2%2B2z%2B1%3D0%5C%5CD%3Db%5E2-4ac%3D2%5E2-4%5Ccdot3%5Ccdot1%3D4-12%3D-8%5C+%5Ctextless+%5C+0)
![D\ \textless \ 0](https://tex.z-dn.net/?f=D%5C+%5Ctextless+%5C+0)
, уравнение действительных корней не имеет
![z= \sqrt{2} t\\ t= \frac{1}{ \sqrt{2} }](https://tex.z-dn.net/?f=z%3D+%5Csqrt%7B2%7D+t%5C%5C+t%3D+%5Cfrac%7B1%7D%7B+%5Csqrt%7B2%7D+%7D+)
Возвращаемся к замене
Ответ: ![(-1)^k\cdot \frac{1}{4} +k,k \in Z](https://tex.z-dn.net/?f=%28-1%29%5Ek%5Ccdot+%5Cfrac%7B1%7D%7B4%7D+%2Bk%2Ck+%5Cin+Z)