Question Number 5556 by Rasheed Soomro last updated on 20/May/16
$$\mathrm{The}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{diretly}\: \\ $$$$\mathrm{proportional}\:\mathrm{to}\:\mathrm{the}\:\mathrm{square}\:\mathrm{of} \\ $$$$\mathrm{its}\:\mathrm{diameter}. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{constant}\:\mathrm{of}\:\mathrm{proportionality}? \\ $$
Commented by Yozzii last updated on 20/May/16
$${A}\propto{d}^{\mathrm{2}} \\ $$$${A}={k}\left(\mathrm{2}{r}\right)^{\mathrm{2}} \:\:\:\left({d}=\mathrm{2}{r}>\mathrm{0}\right) \\ $$$$\pi{r}^{\mathrm{2}} =\mathrm{4}{kr}^{\mathrm{2}} \\ $$$$\Rightarrow{k}=\frac{\pi}{\mathrm{4}}\:. \\ $$$${Alternatively},\:{since}\:{A}=\frac{\pi}{\mathrm{4}}{d}^{\mathrm{2}} \\ $$$$\Rightarrow{A}\propto{d}^{\mathrm{2}} \Rightarrow{k}=\pi/\mathrm{4} \\ $$
Commented by Rasheed Soomro last updated on 20/May/16
$$\mathrm{Your}\:\mathrm{way}\:\mathrm{is}\:\mathrm{better}\:\mathrm{than}\:\mathrm{the}\:\mathrm{way} \\ $$$$\mathrm{I}\:\mathrm{saw}\:\mathrm{in}\:\mathrm{a}\:\mathrm{book}! \\ $$