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Question Number 164347 by ajfour last updated on 16/Jan/22

Commented by ajfour last updated on 16/Jan/22

$$FindmaximumcoloredareaA(a,b).$$

Commented by mr W last updated on 16/Jan/22

Commented by mr W last updated on 16/Jan/22

$$A=A_1+A_2$$$$A_1=\frac{ab}{2}isfixed.$$$$A_{max}\iff A_{2,max}$$$$duetosymmetry,A_{2,max}iswhenc$$$$ishorizontal.$$$$c=\sqrt{a^2+b^2}$$$$A_{2,max}=\frac{2}{3}\times c\times {\left(\frac{c}{2}\right)}^2=\frac{c^3}{6}=\frac{{(a^2+b^2)}^{\frac{3}{2}}}{6}$$$$\Rightarrow A_{max}=\frac{ab}{2}+\frac{{(a^2+b^2)}^{\frac{3}{2}}}{6}$$

Commented by ajfour last updated on 16/Jan/22

$$Thankssir.$$

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