heureka

help please!!!

$\dfrac{2027!+2028!}{2028!-2029!}$

$\begin{array}{|rcll|} \hline && \dfrac{2027!+2028!}{2028!-2029!} \\\\ &=& \dfrac {\frac{2028!}{2028} +2028!} {2028!-2028!\cdot 2029} \\\\ &=& \dfrac { 2028!\cdot \left( \frac{1}{2028} + 1\right) } { 2028!\cdot (1-2029) } \\\\ &=& \dfrac { \frac{1}{2028} + 1} { 1-2029 } \\\\ &=& \dfrac { \frac{1}{2028} + 1} { -2028 } \\\\ &=& \dfrac { 1+2028 } { -2028^2 } \\\\ &=& -\dfrac { 2029 } { 2028^2 } \\\\ \hline \end{array}$

find the equation of a line, in general form that is parallel to hte line 3x-5y-15=0 and passes threw the point (3,-2)

Formula:

$\begin{array}{|rcll|} \hline ax+by+c &=& 0 \quad & | \quad \qquad \text{parallel and passes through the point } P\binom{x_y}{y_p} \\\\ ax+by-(ax_p+by_p) &=& 0 \\ \hline \end{array}$

The equation:

$\begin{array}{|rcll|} \hline 3x-5y-15 &=& 0\quad & | \quad a=3 \quad b= -5 \\\\ ax+by-(ax_p+by_p) &=& 0 \quad & |\quad x_p = 3 \quad y_p = -2 \\ 3x-5y-[3\cdot 3-5\cdot (-2)] &=& 0 \\ 3x-5y-(9+10) &=& 0 \\ \mathbf{3x-5y-19} & \mathbf{=} & \mathbf{0} \qquad \text{or } \qquad \mathbf{y = \frac35\cdot x -\frac{19}{5} }\\ \hline \end{array}$

|3x-1|=|2x+6|

$\begin{array}{|rcll|} \hline |3x-1| &=& |2x+6| \quad & | \quad \text{square both sides} \\ (3x-1)^2 &=& (2x+6)^2 \\ 9x^2-6x+1 &=& 4x^2+24x+36 \\ 9x^2-4x^2-6x-24x+1-36 &=& 0 \\ 5x^2-30x-35 &=& 0 \\ 5x^2-30x-35 &=& 0 \quad & | \quad : 5 \\ x^2-6x-7 &=& 0 \\ (x-7)(x+1) &=& 0 \\ \hline \end{array}$

x = 7 or x = -1

check:

$\begin{array}{|rcll|} \hline |3x-1| &=& |2x+6| \quad & | \quad x=-1 \\ |3\cdot(-1)-1| &=& |2\cdot(-1)+6| \\ |-4| &=& |4| \\ 4 &=& 4 \ \checkmark \\\\ |3x-1| &=& |2x+6| \quad & | \quad x=7 \\ |3\cdot7-1| &=& |2\cdot7+6| \\ |20| &=& |20| \\ 20 &=& 20 \ \checkmark \\ \hline \end{array}$

what is the vertex of this equation 3x^2-11x-4=y

Formula:

$\begin{array}{|rcll|} \hline ax^2+bx+c &=& 0 \\\\ x_v &=& -\frac{b}{2a} \\ y_v &=& c - \frac{b^2}{4a} \\ \hline \end{array}$

Vertex:

$\begin{array}{|rcll|} \hline y &=& 3x^2-11x-4 \quad & | \qquad a = 3 \qquad b=-11 \qquad c = -4 \\\\ x_v &=& -\frac{(-11)}{2\cdot 3} \\ &=& \frac{11}{6} \\ \mathbf{x_v} &\mathbf{=}& \mathbf{1.8\bar{3}} \\\\ y_v &=& -4 - \frac{(-11)^2}{4\cdot 3} \\ &=& -4 - \frac{121}{12} \\ &=& -\frac{169}{12} \\ \mathbf{y_v} &\mathbf{=}& \mathbf{-14.08\bar{3}} \\ \hline \end{array}$

unvereinbare Einheiten

A barn is 100 ft long and 40 feet wide. A cross section of the roof is the inverted catenary

y=31-10(e^x/20+e^-x/20). Find the number of square feet of roofing of the barn.

I assume roofing of the barn is length of catenary  * wide of the barn:

$\begin{array}{|rcll|} \hline f(x) = y &=& 31-10\cdot \left( e^{\frac{x}{20}} + e^{-\frac{x}{20}} \right) \quad & |\quad \left( e^{\frac{x}{20}} + e^{-\frac{x}{20}} \right) = 2\cdot \cosh(\frac{x}{20}) \\ &=& 31-10\cdot \left[ 2\cdot \cosh(\frac{x}{20}) \right] \\ f(x) &=& 31-20\cdot \cosh(\frac{x}{20}) \\\\ f'(x) &=& -20\cdot \sinh(\frac{x}{20}) \cdot \frac{1}{20} \\ f'(x) &=& - \sinh(\frac{x}{20}) \\\\ [f'(x)]^2 &=& \sinh^2(\frac{x}{20}) \\ \hline \end{array}$

The graph of the function f(x):

The length L of the catenary is:

$\begin{array}{|rcll|} \hline L &=& \int \limits_{-50}^{50} \sqrt{1+[f'(x)]^2}\ dx \\ &=& \int \limits_{-50}^{50} \sqrt{1+\sinh^2(\frac{x}{20})}\ dx \quad & |\quad 1+\sinh^2(\frac{x}{20}) = \cosh^2(\frac{x}{20}) \\ &=& \int \limits_{-50}^{50} \sqrt{\cosh^2(\frac{x}{20})}\ dx \\ &=& \int \limits_{-50}^{50} \cosh(\frac{x}{20})\ dx \\ &=& [~ 20\cdot \sinh(\frac{x}{20}) ~]_{-50}^{50} \\ &=& 20\cdot [~ \sinh(\frac{x}{20}) ~]_{-50}^{50} \\ &=& 20\cdot [~ \sinh(\frac{50}{20})-\sinh(\frac{-50}{20}) ~]\\ &=& 20\cdot [~ \sinh(2.5)-\sinh(-2.5) ~]\\ &=& 20\cdot [~ \sinh(2.5)+\sinh(2.5) ~]\\ &=& 20\cdot 2\cdot \sinh(2.5) \\ &=& 40\cdot \sinh(2.5) \\ &=& 40\cdot 6.0502044810397875 \\ &=&\mathbf{ 242.0081792415915\ feet} \\ \hline \end{array}$

The number of square feet of roofing of the barn is:

$\begin{array}{|rcll|} \hline && 242.0081792415915\ feet \times 40 feet \\ &=& \mathbf{9680.32716964\ square\ feet} \\ \hline \end{array}$

96/35=2.74285714286 

x^2*(-0,11 * e^-0,11x) = ?

$\begin{array}{|rcll|} \hline x^2 &=& e^{\ln(x^2)} \\ x^2 &=& e^{2\cdot\ln(x)} \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline && x^2*(-0,11 * e^{-0,11x}) \\ &=& -0,11 * x^2 * e^{-0,11x} \quad & | \quad x^2 = e^{2\cdot\ln(x)}\\ &=& -0,11 * e^{2\cdot\ln(x)}* e^{-0,11x} \\ &=& -0,11 * e^{2\cdot\ln(x)-0,11x} \\ \hline \end{array}$

how to graph 5x-7y=10

9= 5+V         V=...?

V = 9 - 5

V = 4