heureka

tangent condition: $$7.75^2+4^2=8.75^2$$

$$7.75^2+4^2\stackrel{?}{=}8.75^2$$

$$(\frac{31}{4})^2 +(4)^2\stackrel{?}{=}(\frac{35}{4})^2 \quad | \quad *4^2$$

$$31^2+(4^2)^2 \stackrel{?}{=}35^2$$

$$31^2+16^2 \stackrel{?}{=}35^2$$

$$961+256 \stackrel{?}{=}1225$$

$$1217\neq1225$$    not a tangent!

$$\dfrac{L}{\sqrt{L}}*\sqrt{(\sqrt{L}+\sqrt{K})^2-x^2}$$

$$L-K\neq\left( \sqrt{L}+\sqrt{K} \right)^2$$

$$3d\equiv 1 \bmod 1288 \Rightarrow 3d =1288m+1 \Rightarrow 1288m+1 \mbox{ must be divisible by 3 without rest!}$$

$$\Rightarrow 1288m \equiv 2 \bmod 3$$

$$\text{here } \quad 1288\equiv 1 \bmod 3 \quad \mbox{ and } \quad \textcolor[rgb]{1,0,0}{2}*1288\equiv \textcolor[rgb]{1,0,0}{2} \bmod 3$$

$$\mbox{so }\quad m=2 \quad \mbox{ and } \quad d=\frac{1288*2+1}{3}=\frac{2577}{3}=859$$

$$\boxed{3*\textcolor[rgb]{1,0,0}{859}\equiv 1 \bmod 1288}$$

  $$x^{ (\frac{7}{3}) }=x^{2.\bar{3}}$$

  $$\boxed{\mbox{Surface area of a sphere: } \quad S=\pi d^2}$$

$$S_{Titan}=\pi d_{Titan}^2$$

$$S_{Rhea}=\pi d_{Rhea}^2$$

$$\dfrac{S_{Titan}} {S_{Rhea}} = \dfrac{\not{\pi} d_{Titan}^2} {\not{\pi} d_{Rhea}^2} =\dfrac{34}{3}$$

$$\dfrac{d_{Titan}^2} {d_{Rhea}^2} =\dfrac{34}{3}$$

$$d_{Titan}^2=d_{Rhea}^2*\left(\frac{34}{3} \right)$$

$$d_{Titan}=d_{Rhea}*\sqrt{\left(\frac{34}{3} \right)} \quad | \quad d_{Rhea}=1530\;km$$

$$\textstyle{d_{Titan}=1530\;km*\sqrt{\left(\frac{34}{3} \right)} =1530\;km*3.36650164612 = 5150.75\;km\approx5151\;km}$$

Diameter of Titan is 5151 km

$$h=\dfrac{\sin{(41\ensuremath{^\circ} )}*\sin{(48\ensuremath{^\circ})}}{\sin{(41\ensuremath{^\circ}+48\ensuremath{^\circ})}}*140\;m = 68.27\;m$$

What was the total length of wire used for the cuboid ?

length = 4a+4b+4c = 4 * 15cm + 4 * 11cm + 4 * 30cm = 4 * (15cm + 11cm + 30cm) = 4*56cm = 224cm

$$\underbrace{5}_{a=5}x^2\underbrace{-3}_{b=-3}x\underbrace{+1}_{c=1}=0 \quad ?$$

$$discriminant = b^2-4ac =(-3)^2-4*5*1=9-20=-11$$

$$\textcolor[rgb]{1,0,0}{discriminant < 0\qquad \mbox{no real solution}}$$

$$\\A=\dfrac{1}{2}*5*8*(\dfrac{1}{2}\sqrt{3})\\ A=10\sqrt{3}=17.32$$