heureka

If the least common multiple of A and B is 1575, and the ratio A of B to 3 is 7 ,

then what is their greatest common divisor?

$\begin{array}{rcll} \hline lcm(a,b) = 1575 &=& 3^2\times 5^2\times 7 \\ \hline \end{array}$

$\begin{array}{rcll} a &=& 3^2\times 5^2 \\ b &=& 3\times 5^2 \times 7 \\\\ \dfrac{a}{b} = \dfrac{3^2\times 5^2}{3\times 5^2 \times 7 } &=& \dfrac{3}{7}\\ \end{array}$

$\begin{array}{rcll} gcd(a,b) &=& 3\times 5^2 \times 7^0 = 75 \\ \end{array}$

The same question:

Suppose that b  is a positive integer greater than or equal to 2. 

When 197 is converted to base b, the resulting representation has 4 digits.

What is the number of possible values for b?

$\begin{array}{|rcll|} \hline 197_{10} &=& 11000101_2 \\ 197_{10} &=& 21022_3 \\ 197_{10} &=& 3011_{\color{red}4} \\ 197_{10} &=& 1242_{\color{red}5} \\ 197_{10} &=& 525_6 \\ 197_{10} &=& 401_7 \\ 197_{10} &=& 305_8 \\ 197_{10} &=& 238_9 \\ 197_{10} &=& 197_{10} \\ \cdots \\ \hline \end{array}$

b is 4 or 5

If sin(A) = -0.4382.  0 < A < 360.

Find two possible values for A.

$\begin{array}{|rcll|} \hline \sin(A) &=& -0.4382 \\ A_1 &=& \arcsin(-0.4382) + z\cdot 360^{\circ} \quad & | \quad z \in \mathbb{Z} \\ A_1 &=& -25.9890903445^{\circ} + z\cdot 360^{\circ} \\ A_1 &=& -25.9890903445^{\circ}+360^{\circ} + z\cdot 360^{\circ} \\ \mathbf{A_1} &\mathbf{=}& \mathbf{334.010909656^{\circ} + z\cdot 360^{\circ}} \\\\ \sin(A)=\sin(180^{\circ}-A) &=&-0.4382 \\ 180^{\circ}-A_2 &=& \arcsin(-0.4382) + z\cdot 360^{\circ} \quad & | \quad z \in Z \\ A_2 &=& 180^{\circ}- \arcsin(-0.4382) + z\cdot 360^{\circ} \\ A_2 &=& 180^{\circ}- (-25.9890903445^{\circ}) + z\cdot 360^{\circ} \\ A_2 &=& 180^{\circ}+25.9890903445^{\circ}) + z\cdot 360^{\circ} \\ \mathbf{A_2} &\mathbf{=}&\mathbf{205.989090344^{\circ}+ z\cdot 360^{\circ}} \\ \hline \end{array}$

0 < A < 360.

$\begin{array}{|rcll|} \hline \mathbf{A_1} &\mathbf{=}& \mathbf{334.010909656^{\circ} } \\ \mathbf{A_2} &\mathbf{=}& \mathbf{205.989090344^{\circ}} \\ \hline \end{array}$

If 25.12 is the area of a circle, what is the circumference?

Let A = 25.12 the area of a circle

Let C = circumference of a circle

1. Radius r = ?

$\begin{array}{|rcll|} \hline A &=& \pi r^2 \\ \frac{A}{\pi} &=& r^2 \\ \mathbf{\sqrt{\frac{A}{\pi} } } & \mathbf{=} & \mathbf{r} \\ \hline \end{array}$

2. Circumference

$\begin{array}{|rcll|} \hline C &=& 2\pi r \quad &| \quad r=\sqrt{\frac{A}{\pi}} \\ C &=& 2\pi\sqrt{\frac{A}{\pi}} \\ C &=& 2\sqrt{\pi^2\frac{A}{\pi}} \\ \mathbf{C }& \mathbf{=} & \mathbf{2\sqrt{A\cdot \pi }} \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline \mathbf{C }& \mathbf{=} & \mathbf{2\sqrt{A\cdot \pi }} \quad & | \quad A = 25.12 \\ C &=& 2\sqrt{25.12\cdot 3.14159265359} \\ C &=& 2\sqrt{78.9168074582} \\ C &=& 2\cdot 8.88351323848 \\ C &=& 17.7670264770 \\ \hline \end{array}$

A circle of radius 5 with its center at (0,0) is drawn on a Cartesian coordinate system.

How many lattice points (points with integer coordinates) lie within or on this circle?

Let K the circle with center (0,0)

Let r  the radius of the circle.

Let g(r) are the lattice points lie within or on this circle. $[~g(r) :=|{(x,y)\in \mathbb{Z^2}|x^2+y^2\le r^2}|~]$

We have four subsets A, B, C, and D with g(r) = A+B+C+D

• A = {(0,0)} = 1   
• B = Cut K with axis without (0,0) $= 2\cdot r + 2\cdot r = 4\cdot r$ 
• $C_1, C_2, C_3, C_4 =$ squares parallel to the axis with one corner in (0,0) and edge length $\frac{r}{\sqrt2}$   
  $= 4\cdot \left( \left[ \dfrac{r}{\sqrt2} \right] \right)^2$   with $[x] =$ greatest integer number   
•  D = remain

Subset D:

For D there is no Formula in close form, but though we have:

$\frac{D}{8} = [\sqrt{r^2 - a^2}] +[\sqrt{r^2 - (a+1)^2}] + \cdots + [\sqrt{r^2 - (r-1)^2}]$ with $a =\left[ \dfrac{r}{\sqrt2} \right] + 1$  and 

$[x] =$  greatest integer number

g(5) = ?

$\begin{array}{|rcll|} \hline A &=& 1 \\ B &=& 4\cdot 5 \\ &=& 20 \\ C &=& 9\cdot 4 \quad &| \quad \left[ \dfrac{r}{\sqrt2} \right] = \left[ \dfrac{5}{\sqrt2} \right] = 3 \\ &=& 36 \\ D &=& 8\cdot ( [\sqrt{5^2 - 4^2}] ) \quad &| \quad a =\left[ \dfrac{5}{\sqrt2} \right] + 1 = 3 +1 = 4 \qquad r-1 = 5-1 = 4 \\ &=& 8\cdot ( [\sqrt{25 - 16}] ) \\ &=& 8\cdot ( [\sqrt{9}] ) \\ &=& 8\cdot [3] \\ &=& 8\cdot 3 \\ &=&24\\ \hline \end{array}$

$\begin{array}{|rcll|} \hline g(5) &=& A+B+C+D \\ &=& 1 + 20 + 36 + 24 \\ &=& 81 \\ \hline \end{array}$

There are 81 lattice points.

Ich habe eine Liter Flasche mit 57% Alkohol. Diesen möchte ich auf 42% Alkohol reduzieren.

Wieviel Wasser muss ich pro liter hinzufügen?

57 % von 1 Liter sind 570 ml Alkohol.

Diese 570ml Alkohol sind immer noch nach dem Hinzufügen von Wasser enthalten.

Sie entsprechen jetzt aber 42 %.

Somit: 42 % von x Liter = 570 ml Alkohol.

$x=\frac{570}{\frac{42}{100}} \\ x=\frac{570\cdot 100}{42} \\ x=1357.14285714\ ml$

Wir hatten ein Liter = 1000 ml

jetzt haben wir 1357.14285714 ml

also wurden 1357.14285714 ml - 1000 ml = 357.142857143 ml Wasser hinzugefügt.

How tall is a bridge if a 6-foot-tall person standing 100 feet away

can see the top of the bridge at an angle of 20 degrees to the horizon?

$Let\ x = \text{ height of the bridge}$

$\begin{array}{|rcll|} \hline \tan(20^{\circ}) &=& \frac{x-6 }{100 } \\ 100\cdot \tan(20^{\circ}) &=& x-6 \\ 6 + 100\cdot \tan(20^{\circ}) &=& x \\ x &=& 6 + 100\cdot \tan(20^{\circ}) \\ x &=& 6 + 100\cdot 0.36397023427 \\ x &=& 6 + 36.3970234266 \\ \mathbf{x} & \mathbf{=} & \mathbf{42.3970234266\ \text{feet}} \\ \hline \end{array}$

simplfy this expression. sqrt3(((8*x^3)*yx^6)*z^3)

$\begin{array}{|rcll|} \hline && \sqrt{3}\cdot \{~[(8\cdot x^3)\cdot yx^6]\cdot z^3~\} \\ &=& \sqrt{3}\cdot 8 \cdot x^9 \cdot y \cdot z^3 \\ \hline \end{array}$

Light of wavelength 619 nm passes through a double slit and makes its first max (m=1) at ab abgke if 0.226 deg. What is the separation of the slits, d, IN MILLIMETERS? Rearrange and solve, please. 

$d\sin(\theta)= m\lambda$

$\begin{array}{rcll} Let\ \lambda &=& 619\ nm \\ \lambda &=& 619 \cdot 10^{-9}\ m \\ \lambda &=& 619 \cdot 10^{-6}\ mm \\\\ Let\ \theta &=& 0.226^{\circ} \\ \end{array}$

$\begin{array}{|rcll|} \hline d\sin(\theta) &=& m \lambda \\ d &=& m\cdot \frac{\lambda}{\sin(\theta)} \quad & | \quad m=1 \\ d &=& 1\cdot \frac{619 \cdot 10^{-6}\ mm }{\sin(0.226^{\circ})} \\ d &=& 156929.997726 \cdot 10^{-6} \ mm \\ d &=& 0.156929997726 \cdot 10^{6}\cdot 10^{-6} \ mm \\ d &=& 0.156929997726 \ mm \\ \hline \end{array}$