A)
3x-7=65
3x=65+7
3x=72
x=72:3
x=24
б)
6x-x=35
5x=35
x=35:5
x=7
в)
c•12•8=960
c•96=960
c=960:96
c=10
a)
7x+12+4x=11x+12
100 \ 5 = 20 % цветов приходится на одну часть
20 * 2 = 40% - розы
20 * 3 = 60 % другие цветы
Найдем плотность распределения
![f(x)](https://tex.z-dn.net/?f=f%28x%29)
, как производную от функции распределения
![F(x)](https://tex.z-dn.net/?f=F%28x%29)
(это само определение плотности):
![\displaystyle f(x)= \dfrac{d}{dx} F(x)= \frac{d}{dx} (tgx)= \frac{1}{\cos^2x} = \frac{\sin^2x+\cos^2x}{\cos^2x}=tg^2x+1](https://tex.z-dn.net/?f=%5Cdisplaystyle+f%28x%29%3D+%5Cdfrac%7Bd%7D%7Bdx%7D+F%28x%29%3D+%5Cfrac%7Bd%7D%7Bdx%7D+%28tgx%29%3D+%5Cfrac%7B1%7D%7B%5Ccos%5E2x%7D+%3D+%5Cfrac%7B%5Csin%5E2x%2B%5Ccos%5E2x%7D%7B%5Ccos%5E2x%7D%3Dtg%5E2x%2B1)
![f(x)=\begin{cases} & \text{ } 0,~~~~~ ~~~~~~~~~~~~~~x\ \textless \ 0 \\ & \text{ } tg^2x+1,~~~~~0\ \textless \ x\ \textless \ \frac{\pi}{4} \\ & \text{ } 0 ,~~~~~~~~~~~~~~~~~~~x\ \textgreater \ \frac{\pi}{4} \end{cases}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cbegin%7Bcases%7D%0A+%26+%5Ctext%7B++%7D+0%2C~~~~~+~~~~~~~~~~~~~~x%5C+%5Ctextless+%5C+0+%5C%5C+%0A+%26+%5Ctext%7B++%7D+tg%5E2x%2B1%2C~~~~~0%5C+%5Ctextless+%5C+x%5C+%5Ctextless+%5C++%5Cfrac%7B%5Cpi%7D%7B4%7D++%5C%5C+%0A+%26+%5Ctext%7B++%7D+0+%2C~~~~~~~~~~~~~~~~~~~x%5C+%5Ctextgreater+%5C+%5Cfrac%7B%5Cpi%7D%7B4%7D+%0A%5Cend%7Bcases%7D)
- плотность распределения.
б) Вычислить математическое ожидание M(x)
![\displaystyle M(x)= \int\limits^{\frac{\pi}{4} }_0 {xf(x)} \, dx =\int\limits^{\frac{\pi}{4} }_0(xtg^2x+x)dx=x\cdot tgx\bigg|^{ \frac{\pi}{4} }_0-\\ \\ - \frac{1}{2}\ln|tg^2x+1|\bigg|^{\frac{\pi}{4} } _0\approx0.439](https://tex.z-dn.net/?f=%5Cdisplaystyle+M%28x%29%3D+%5Cint%5Climits%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D+%7D_0+%7Bxf%28x%29%7D+%5C%2C+dx+%3D%5Cint%5Climits%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D+%7D_0%28xtg%5E2x%2Bx%29dx%3Dx%5Ccdot+tgx%5Cbigg%7C%5E%7B+%5Cfrac%7B%5Cpi%7D%7B4%7D+%7D_0-%5C%5C+%5C%5C+-+%5Cfrac%7B1%7D%7B2%7D%5Cln%7Ctg%5E2x%2B1%7C%5Cbigg%7C%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D+%7D+_0%5Capprox0.439)
в) Дисперсия D(x)
![D(x)=\displaystyle \int\limits^{\frac{\pi}{4} }_0x^2f(x)dx= \int\limits^{\frac{\pi}{4} }_0(x^2tg^2x+x^2)dx\approx0.053](https://tex.z-dn.net/?f=D%28x%29%3D%5Cdisplaystyle+%5Cint%5Climits%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D+%7D_0x%5E2f%28x%29dx%3D+%5Cint%5Climits%5E%7B%5Cfrac%7B%5Cpi%7D%7B4%7D+%7D_0%28x%5E2tg%5E2x%2Bx%5E2%29dx%5Capprox0.053)
г)
![\sigma(x)= \sqrt{D(x)}\approx 0.23](https://tex.z-dn.net/?f=%5Csigma%28x%29%3D+%5Csqrt%7BD%28x%29%7D%5Capprox+0.23)