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Question Number 181783 by mr W last updated on 30/Nov/22

$$onesideofatriangleis20cm.the$$$$othertwosidesareinratio1:3.$$$$1)whatisthemaximumareaofthe$$$$triangle,ifexists?$$$$2)whatisthemaximunperimeter$$$$ofthetriangle,ifexists?$$

Answered by Frix last updated on 30/Nov/22

$$\mathrm{Let}\mathrm{the}\mathrm{sides}x,3x$$$$5<x<10$$$$\mathrm{The}\mathrm{collapsed}\mathrm{triangle}10,20,30\mathrm{has}\mathrm{the}$$$$\mathrm{maximum}\mathrm{perimeter}60$$$$\mathrm{The}\mathrm{area}\mathrm{is}A\left(x\right)=2\sqrt{-x^4+125x^2-2500}$$$$A^{\prime }\left(x\right)=\frac{-4x(2x^2-125)}{A\left(x\right)}=0\wedge 5<x<10\Rightarrow x=\frac{5\sqrt{10}}{2}$$$$\Rightarrow \mathrm{maximum}\mathrm{area}\mathrm{is}75$$

Commented by mr W last updated on 30/Nov/22

$$thankssir!$$

Answered by MJS_new last updated on 30/Nov/22

$$\mathrm{generally}\mathrm{for}\mathrm{a}\mathrm{given}\mathrm{side}a\mathrm{and}\mathrm{a}\mathrm{ratio}$$$$b:c=1:v\mathrm{with}v>1\mathrm{we}\mathrm{get}\mathrm{the}\mathrm{maximum}$$$$\mathrm{area}$$$$\frac{a^{2}v}{2(v^2-1)}\mathrm{with}b=\frac{\sqrt{v^2+1}}{v^2-1}a\mathrm{and}c=vb$$

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