Question Number 49360 by behi83417@gmail.com last updated on 06/Dec/18
Commented by behi83417@gmail.com last updated on 06/Dec/18
$$\boldsymbol{\mathrm{when}}\:\boldsymbol{\mathrm{area}}\:\boldsymbol{\mathrm{of}}:{AGEC},\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{maximum}}, \\ $$$$\boldsymbol{\mathrm{then}}:\:\:\frac{\boldsymbol{\mathrm{GE}}}{\boldsymbol{\mathrm{AC}}}=? \\ $$
Commented by MJS last updated on 06/Dec/18
![I think if you don′t fix D ⇒ ∣AD∣=∞; AC∥GE ((GE)/(AC))=(1/2) because greatest trapezoid can be found as follows: ((AC)/2)=r ((GE)/2)=rcos α h=rsin α area(ACEG)=r^2 (1+cos α)sin α (d/dα)[r^2 (1+cos α)sin α]=0 r^2 (2cos^2 α +cos α −1)=0 cos α =(1/2) ⇒ ((GE)/(AC))=(1/2)](https://www.tinkutara.com/question/Q49369.png)
$$\mathrm{I}\:\mathrm{think}\:\mathrm{if}\:\mathrm{you}\:\mathrm{don}'\mathrm{t}\:\mathrm{fix}\:{D}\:\Rightarrow\:\mid{AD}\mid=\infty;\:{AC}\parallel{GE} \\ $$$$\frac{{GE}}{{AC}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{because}\:\mathrm{greatest}\:\mathrm{trapezoid}\:\mathrm{can}\:\mathrm{be}\:\mathrm{found}\:\mathrm{as} \\ $$$$\mathrm{follows}: \\ $$$$\frac{{AC}}{\mathrm{2}}={r} \\ $$$$\frac{{GE}}{\mathrm{2}}={r}\mathrm{cos}\:\alpha \\ $$$${h}={r}\mathrm{sin}\:\alpha \\ $$$$\mathrm{area}\left({ACEG}\right)={r}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{cos}\:\alpha\right)\mathrm{sin}\:\alpha \\ $$$$\frac{{d}}{{d}\alpha}\left[{r}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{cos}\:\alpha\right)\mathrm{sin}\:\alpha\right]=\mathrm{0} \\ $$$${r}^{\mathrm{2}} \left(\mathrm{2cos}^{\mathrm{2}} \:\alpha\:+\mathrm{cos}\:\alpha\:−\mathrm{1}\right)=\mathrm{0} \\ $$$$\mathrm{cos}\:\alpha\:=\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\:\frac{{GE}}{{AC}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by behi83417@gmail.com last updated on 06/Dec/18
$${thank}\:{you}\:{sir}\:{for}\:{attention}.{this}\:{is}\:{my} \\ $$$${typo}.{excuse}\:{me}. \\ $$$${yes},\boldsymbol{\mathrm{D}}\:{is}\:{a}\:{fix}\:{point}. \\ $$
Commented by peter frank last updated on 06/Dec/18
$$\mathrm{sir}\:\mathrm{plz}\:\:\mathrm{check}\:\mathrm{QN}\:\:\mathrm{49357},\mathrm{49358} \\ $$