Alan

$$$$f(x)=(5x-4)^\frac{1}{2}$$
$$f'(x)=\frac{1}{2}(5x-4)^{-\frac{1}{2}}*5=\frac{5}{2(5x-4)^{\frac{1}{2}}}$$
$$f'(4)=\frac{5}{2(5*4-4)^{\frac{1}{2}}}=\frac{5}{2(20-4)^\frac{1}{2}}$$
$$=\frac{5}{2*\sqrt{16}}=\frac{5}{2*4}=\frac{5}{8}$$

The chance is 5*10⁶/7*10⁹ or 5/7000.  That is, there are 5 chances in 7000 of being born in that nation (or 1 in 1400 if you prefer).  

If you've two sides and the included angle use the Cosine rule.

If you've three sides use the Cosine rule.

If you've two angles and a side use the Sine rule.

In the case when there is an even number of numbers it is normal to take the median as the average of the two nearest the middle.  So, in this case the median would be (33+43)/2 = 38

model nucleus/model atom = actual nucleus/actual atom

n/5 = 10^-15/10^-10

n/5 = 10^-15*10⁺¹⁰ = 10^-15+10 = 10^-5

Multiply both sides by 5

n = 5*10^-5 cm

n = 0.00005 cm!  She needs a bigger model!

You can simplify it as follows (where the capital pi symbol means take the product of what follows).  However, the site calculator doesn't have this product function as far as I can tell.

b = 72/8 cm = 9cm

tan(22.5°)=(b/2)/h  so h = (b/2)/tan(22.5°) cm

Area of 1/8 of Octagon = (b/2)*h = (b/2)²/tan(22.5°)

Area of Octagon = 8*(b/2)²/tan(22.5°)= 2b²/tan(22.5°)

$${\mathtt{Area}} = {\frac{{\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{9}}}^{{\mathtt{2}}}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{22.5}}^\circ\right)}}} \Rightarrow {\mathtt{Area}} = {\mathtt{391.102\: \!597\: \!104\: \!531\: \!143\: \!5}}$$

Area ≈ 391.1 cm²

$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{7}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{53}} = {\frac{{\mathtt{11}}}{{\mathtt{3}}}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{1\,335}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,\small\textbf+\,}}{\mathtt{21}}\right)}{{\mathtt{6}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{1\,335}}}}{\mathtt{\,\times\,}}{i}{\mathtt{\,-\,}}{\mathtt{21}}\right)}{{\mathtt{6}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}\left({\frac{{\mathtt{7}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6.089\: \!608\: \!635\: \!484\: \!241\: \!2}}{i}\right)\\
{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{{\mathtt{7}}}{{\mathtt{2}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6.089\: \!608\: \!635\: \!484\: \!241\: \!2}}{i}\\
\end{array} \right\}$$

slope = (y2 - y1)/(x2 - x1)

slope = (-11 - 4)/(-3 - 2) = -15/(-5) = +3

$$$$\frac{\sqrt{32}}{2}=\frac{\sqrt{16*2}}{2}=\frac{4\sqrt2}{2}=2\sqrt2$$

Since √2 < 2 then 2√2 < 4

Also: nice piece of reasoning by Goldenleaf!