Question Number 207712 by efronzo1 last updated on 24/May/24
Answered by Frix last updated on 24/May/24
![For x=q and A= ((p),(0) ) the area is (q−p)(√p) ((d[(q−p)(√p)])/dp)=0 ((q−3p)/(2(√p)))=0 ⇒ p=(q/3) Max area =((2(√3)q^(3/2) )/9) q=4 ⇒ answer is ((16(√3))/9)](https://www.tinkutara.com/question/Q207714.png)
$$\mathrm{For}\:{x}={q}\:\mathrm{and}\:{A}=\begin{pmatrix}{{p}}\\{\mathrm{0}}\end{pmatrix}\:\mathrm{the}\:\mathrm{area}\:\mathrm{is}\:\left({q}−{p}\right)\sqrt{{p}} \\ $$$$\frac{{d}\left[\left({q}−{p}\right)\sqrt{{p}}\right]}{{dp}}=\mathrm{0} \\ $$$$\frac{{q}−\mathrm{3}{p}}{\mathrm{2}\sqrt{{p}}}=\mathrm{0}\:\Rightarrow\:{p}=\frac{{q}}{\mathrm{3}} \\ $$$$\mathrm{Max}\:\mathrm{area}\:=\frac{\mathrm{2}\sqrt{\mathrm{3}}{q}^{\frac{\mathrm{3}}{\mathrm{2}}} }{\mathrm{9}} \\ $$$${q}=\mathrm{4}\:\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\frac{\mathrm{16}\sqrt{\mathrm{3}}}{\mathrm{9}} \\ $$