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Question Number 197670 by mr W last updated on 26/Sep/23

Commented by mr W last updated on 26/Sep/23

$$\text{You can't use 'macro parameter character \#' in math mode}$$

Commented by ajfour last updated on 26/Sep/23

https://youtu.be/Y2LCMHAw3Fw?si=Vt7lXmpu1xLMVuB9

Commented by ajfour last updated on 26/Sep/23

A simple educational video i posted hours back, on dy/dx if y=x and if y= x^2

Commented by mr W last updated on 26/Sep/23

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Answered by mr W last updated on 26/Sep/23

Commented by mr W last updated on 26/Sep/23

$$s=sidelengthof\mathrm{\Delta }ABC$$$$\tan \alpha =\frac{s}{3}\times \frac{\sqrt{3}}{2}\times \frac{1}{s-\frac{s}{3}\times \frac{1}{2}}=\frac{\sqrt{3}}{5}$$$$a=\frac{s}{2}\times \tan \alpha =\frac{s}{2}\times \frac{\sqrt{3}}{5}=\frac{s\sqrt{3}}{10}$$$$b=\frac{s}{2}$$$$\beta =90°-60°=30°$$$$c=b-2a\cos \beta =\frac{s}{2}-2\times \frac{s\sqrt{3}}{10}\times \frac{\sqrt{3}}{2}=\frac{s}{5}$$$$\frac{bluearea}{\mathrm{\Delta }ABC}={\left(\frac{c}{s}\right)}^2={\left(\frac{1}{5}\right)}^2=\frac{1}{25}✓$$

Commented by mr W last updated on 26/Sep/23

$$youarerightsir!$$$$itook\tan \alpha wronglyas\sin \alpha .$$

Commented by ajfour last updated on 26/Sep/23

$$trianglesides6sand2c$$$$\frac{6s}{2}\times \tan 30°=3s\tan \alpha +\frac{c}{\cos 30°}..\left(i\right)$$$$\frac{\sin \alpha }{2s}=\frac{\sin (\alpha +60°)}{6s}$$$$\Rightarrow 3\sin \alpha =\frac{1}{2}\sin \alpha +\frac{\sqrt{3}}{2}\cos \alpha$$$$\Rightarrow \tan \alpha =\frac{\sqrt{3}}{5}$$$$nowfrom\left(i\right)$$$$s\sqrt{3}=\frac{3\sqrt{3}}{5}s+\frac{2c}{\sqrt{3}}$$$$\frac{c}{s}=\frac{3}{5}\Rightarrow \frac{2c}{6s}=\frac{1}{5}$$$$r_{area}=\frac{1}{25}$$

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