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Question Number 209223 by Tawa11 last updated on 04/Jul/24

Answered by efronzo1 last updated on 04/Jul/24

((((a/b))^3 +3((a/b)))/(1+3((a/b))^2 )) = ((63)/(62)) (a/b) = x (x^3 +3x)62 = 63(1+3x^2 ) 62x^3 + 186x −189x^2 −63 = 0 (2x−3)(31x^2 −48x+21)=0 x ⇒ x = (a/b) = (3/2)

$$\:\:\: \\ $$$$ \frac{\left(\frac{{a}}{{b}}\right)^{\mathrm{3}} +\mathrm{3}\left(\frac{{a}}{{b}}\right)}{\mathrm{1}+\mathrm{3}\left(\frac{{a}}{{b}}\right)^{\mathrm{2}} }\:=\:\frac{\mathrm{63}}{\mathrm{62}} \\ $$$$\:\: \frac{{a}}{{b}}\:=\:{x}\: \\ $$$$\:\:\:\left({x}^{\mathrm{3}} \:+\mathrm{3}{x}\right)\mathrm{62}\:=\:\mathrm{63}\left(\mathrm{1}+\mathrm{3}{x}^{\mathrm{2}} \:\right) \\ $$$$\:\:\:\mathrm{62}{x}^{\mathrm{3}} \:+\:\mathrm{186}{x}\:−\mathrm{189}{x}^{\mathrm{2}} −\mathrm{63}\:=\:\mathrm{0} \\ $$$$\:\:\:\left(\mathrm{2}{x}−\mathrm{3}\right)\left(\mathrm{31}{x}^{\mathrm{2}} −\mathrm{48}{x}+\mathrm{21}\right)=\mathrm{0} \\ $$$$\:\:\: {x}\: \Rightarrow\:{x}\:=\:\frac{{a}}{{b}}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\:\: \\ $$

Commented by Tawa11 last updated on 04/Jul/24

Thanks sir. I appreciate.

$$\mathrm{Thanks}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{appreciate}. \\ $$

Commented by Spillover last updated on 05/Jul/24

perfect

$${perfect} \\ $$

Answered by MATHEMATICSAM last updated on 04/Jul/24

((a^3 + 3ab^2 )/(b^3 + 3a^2 b)) = ((63)/(62)) Using comp and div ⇒ ((a^3 + 3ab^2 + 3a^2 b + b^3 )/(a^3 + 3ab^2 − b^3 − 3a^2 b)) = ((63 + 62)/(63 − 62)) = 125 ⇒ (((a + b)^3 )/((a − b)^3 )) = 125 ⇒ ((a + b)/(a − b)) = 5 Using comp and div ⇒ (a/b) = ((5 + 1)/(5 − 1)) = (6/4) = (3/2)

$$\frac{{a}^{\mathrm{3}} \:+\:\mathrm{3}{ab}^{\mathrm{2}} }{{b}^{\mathrm{3}} \:+\:\mathrm{3}{a}^{\mathrm{2}} {b}}\:=\:\frac{\mathrm{63}}{\mathrm{62}} \\ $$$$\mathrm{Using}\:\mathrm{comp}\:\mathrm{and}\:\mathrm{div} \\ $$$$\Rightarrow\:\frac{{a}^{\mathrm{3}} \:+\:\mathrm{3}{ab}^{\mathrm{2}} \:+\:\mathrm{3}{a}^{\mathrm{2}} {b}\:+\:{b}^{\mathrm{3}} }{{a}^{\mathrm{3}} \:+\:\mathrm{3}{ab}^{\mathrm{2}} \:−\:{b}^{\mathrm{3}} \:−\:\mathrm{3}{a}^{\mathrm{2}} {b}}\:=\:\frac{\mathrm{63}\:+\:\mathrm{62}}{\mathrm{63}\:−\:\mathrm{62}}\:=\:\mathrm{125} \\ $$$$\Rightarrow\:\frac{\left({a}\:+\:{b}\right)^{\mathrm{3}} }{\left({a}\:−\:{b}\right)^{\mathrm{3}} }\:=\:\mathrm{125} \\ $$$$\Rightarrow\:\frac{{a}\:+\:{b}}{{a}\:−\:{b}}\:=\:\mathrm{5} \\ $$$$\mathrm{Using}\:\mathrm{comp}\:\mathrm{and}\:\mathrm{div} \\ $$$$\Rightarrow\:\frac{{a}}{{b}}\:=\:\frac{\mathrm{5}\:+\:\mathrm{1}}{\mathrm{5}\:−\:\mathrm{1}}\:=\:\frac{\mathrm{6}}{\mathrm{4}}\:=\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Commented by Tawa11 last updated on 04/Jul/24

Thanks sir. I appreciate.

$$\mathrm{Thanks}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{appreciate}. \\ $$

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