heureka

When a number is divided by 27, it gives a quotient of 218 and a remainder of 14.

What is the number?

\(\begin{array}{|rcll|} \hline \dfrac{x}{27} &=& 218 + \frac{14}{27} \quad & | \quad \cdot 27 \\ x &=& 218\cdot 27 + 14 \\ x &=& 5900 \\ \hline \end{array}\)

13! (13−4)!4!

\(\begin{array}{|rcll|} \hline && \frac{13!}{(13-4)!4!} \\ &=& \dbinom{13}{4} \\ &=& \frac{13}{4}\cdot \frac{12}{3}\cdot \frac{11}{2}\cdot \frac{10}{1} \\ &=& \frac{13}{1}\cdot \frac{12}{12}\cdot 11 \cdot \frac{10}{2} \\ &=& 13\cdot 1\cdot 11\cdot 5 \\ &=& \mathbf{715} \\ \hline \end{array} \)

what is x in sin x =.5

\(\begin{array}{|rcll|} \hline \sin(x) &=& 0.5 \\ x_1 &=& \arcsin(0.5) + z\cdot 360^{\circ} \quad & | \quad z \in Z \\ \mathbf{x_1} &\mathbf{=}& \mathbf{30^{\circ} + z\cdot 360^{\circ}} \\\\ \sin(x)=\sin(180^{\circ}-x) &=& 0.5 \\ 180^{\circ}-x_2 &=& \arcsin(0.5) + z\cdot 360^{\circ} \quad & | \quad z \in Z \\ x_2 &=& 180^{\circ}- \arcsin(0.5) + z\cdot 360^{\circ} \\ x_2 &=& 180^{\circ}- 30^{\circ} + z\cdot 360^{\circ} \\ \mathbf{x_2} &\mathbf{=}&\mathbf{ 150^{\circ} + z\cdot 360^{\circ}} \\ \hline \end{array} \)

One hundred and fifty children attended summer camp.

Seventy-one of the 150 children signed up for swimming and

62 of the 150 children signed up for canoeing.

Twentyeight of the 62 children who signed up for canoeing also signed up for swimming.

Construct a two-way table summarizing the data.

two-way table:

\(\begin{array}{|l|r|r|r|} \hline &\text{Canoeing} &\text{no canoeing}& \text{Total} \\ \hline \text{Swimming} & 28 & 43 & 71 \\ \text{no swimming} & 34 & 45 & 79 \\ \text{Total} & 62 & 88 & 150 \\ \hline \end{array} \)

Venn Diagram:

One hundred and fifty children attended summer camp.

Seventy-one of the 150 children signed up for swimming and

62 of the 150 children signed up for canoeing.

Twentyeight of the 62 children who signed up for canoeing also signed up for swimming.

Construct a two-way table summarizing the data.

\(\begin{array}{|l|r|r|r|} \hline &\text{Canoeing} &\text{no canoeing}& \text{Total} \\ \hline \text{Swimming} & 28 & 43 & 71 \\ \text{no swimming} & 34 & 45 & 79 \\ \text{Total} & 62 & 88 & 150 \\ \hline \end{array} \)

The region bounded by y = 5 and y = x+(4/x) is rotated about the line x=−1.

Find the volume of the resulting solid by any method.

Shell method:

\(\begin{array}{|rcll|} \hline f\left(x\right) &=& x+\frac{4}{x} \\ g\left(x\right) &=& 5 \\ \hline \end{array}\)

Rotation around Vertical line: \(x_\text{axis of rotation}=-1\)

Radius of Rotation: \(r = x-x_\text{axis of rotation} =x-(-1)=x+1\)

Circumference of the cylinder \(= 2\pi r\)

Shell Volume \(= 2\pi r \cdot[ g(x)-f(x)]\ dx\)

Limits of integration:

\(\begin{array}{|rcll|} \hline f\left(x\right) &=& g\left(x\right) \\ x+\frac{4}{x} &=& 5 \quad & | \quad \cdot x \\ x^2+4 &=& 5x \quad & | \quad -5x \\ x^2- 5x +4 &=& 0 \\ (x-1)(x-4) &=& 0 \\ a = 1 &\text{and}& b = 4\\ \hline \end{array} \)

Volume:

\(\begin{array}{|rcll|} \hline V &=& \int \limits_{a}^{b} 2\pi r \cdot[ g(x)-f(x)] \ dx \\ &=& 2\pi \cdot \int \limits_{a}^{b} r \cdot[ g(x)-f(x)] \ dx \\ \hline \end{array} \)

\(\small{ \begin{array}{|rcll|} \hline V &=& 2\pi \cdot \int \limits_{1}^{4} (x+1) \cdot \left[ 5-(x+\frac{4}{x}) \right] \ dx \\ &=& 2\pi \cdot \int \limits_{1}^{4} [ x\cdot( 5-x-\frac{4}{x}) + 5-x-\frac{4}{x} ] \ dx \\ &=& 2\pi \cdot \int \limits_{1}^{4} ( 5x-x^2-4 + 5-x-\frac{4}{x} ) \ dx \\ &=& 2\pi \cdot \int \limits_{1}^{4} ( 4x-x^2+1-\frac{4}{x} ) \ dx \\ &=& 2\pi \cdot [~ 2x^2-\frac{x^3}{3}+x-4\cdot \ln(x) ~]_1^4 \\ &=& 2\pi \cdot \left[~ 2\cdot (4)^2-\frac{(4)^3}{3}+ (4) -4\cdot \ln(4) - \left(2\cdot (1)^2-\frac{(1)^3}{3}+(1)-4\cdot \ln(1) \right) \right]\\ &=& 2\pi \cdot \left[32-\frac{64}{3}+4-4\cdot \ln(4) - \left(2-\frac{1}{3}+1+0 \right) \right] \\ &=& 2\pi \cdot \left[32-\frac{64}{3}+4-4\cdot \ln(4) - 2+\frac{1}{3}-1 \right] \\ &=& 2\pi \cdot \left[33-\frac{63}{3}-4\cdot \ln(4) \right] \\ &=& 2\pi \cdot \left[33-21-4\cdot \ln(2^2) \right] \\ &=& 2\pi \cdot \left[12-4\cdot 2\cdot \ln(2) \right] \\ &=& 2\pi \cdot [12-8\cdot \ln(2) ] \\ &=& 2\pi \cdot 6.45482255552 \\ &=& 40.5568462413 \\ \hline \end{array} } \)

The volume of the resulting solid is 40.5568462413

A map has a scale 6.5 in=32 miles.  How far apart are two cities that are 97.5  inches apart on the map?

\(\begin{array}{|rcll|} \hline && 97.5\ in. \times \frac{32\ \text{miles} }{6.5\ in.} \\ &=& 97.5 \times \frac{32 }{6.5 } \ \text{miles} \\ &=& 480 \ \text{miles} \\ \hline \end{array} \)

The two cities are 480 miles far apart.

Identify the highest common factor of 9 and 30.

\(\begin{array}{lcll} \text{Factors of } 9 &:& 1, {\color{red}3}, 9 \\ \text{Factors of } 30 &:& 1, 2, {\color{red}3}, 5, 6, 10, 15, 30 \\ \end{array} \)

\(\begin{array}{rcll} 3 \text{ is the Greatest Common Factor } \end{array}\)

Wie rechne kann ich aus einer Fläche des Dreiecks von 100 cm2 die Möglichkeit der Seiten ausrechnen ?

Wenn nur die Fläche eines beliebigen Dreiecks gegeben ist, so gibt es unendlich viele Lösungen.

-sinx=12/13

\(\begin{array}{|rcll|} \hline -\sin(x) &=& \frac{12}{13} \\ \sin(x) &=& -\frac{12}{13} \\ x_1 &=& \arcsin(-\frac{12}{13}) + z\cdot 360^{\circ} \quad & | \quad z \in Z \\ \mathbf{x_1} &\mathbf{=}& \mathbf{-67.3801350520^{\circ} + z\cdot 360^{\circ}} \\\\ \sin(x)=\sin(180^{\circ}-x) &=& -\frac{12}{13} \\ 180^{\circ}-x_2 &=& \arcsin(-\frac{12}{13}) + z\cdot 360^{\circ} \quad & | \quad z \in Z \\ x_2 &=& 180^{\circ}- \arcsin(-\frac{12}{13}) + z\cdot 360^{\circ} \\ x_2 &=& 180^{\circ}- (-67.3801350520^{\circ}) + z\cdot 360^{\circ} \\ \mathbf{x_2} &\mathbf{=}&\mathbf{ 247.380135052 + z\cdot 360^{\circ}} \\ \hline \end{array} \)