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Question Number 171437 by solomonwells last updated on 15/Jun/22

make r   the subject of the formula    y=(((pr)/m)  − (p^3 /1))^(−3/2)     find r if y=−8   ,  m=−1,  p=3

$$\mathrm{make}\:\boldsymbol{\mathrm{r}}\:\:\:\mathrm{the}\:\mathrm{subject}\:\mathrm{of}\:\mathrm{the}\:\mathrm{formula} \\ $$$$ \\ $$$$\boldsymbol{\mathrm{y}}=\left(\frac{\boldsymbol{\mathrm{pr}}}{\boldsymbol{\mathrm{m}}}\:\:−\:\frac{\boldsymbol{\mathrm{p}}^{\mathrm{3}} }{\mathrm{1}}\right)^{−\mathrm{3}/\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{find}\:\boldsymbol{\mathrm{r}}\:\mathrm{if}\:\mathrm{y}=−\mathrm{8}\:\:\:,\:\:\mathrm{m}=−\mathrm{1},\:\:\mathrm{p}=\mathrm{3} \\ $$

Answered by alephzero last updated on 15/Jun/22

y = (((pr)/m)−p^3 )^(−3/2)   ⇒ y^(−2/3)  = ((pr)/m)−((mp^3 )/m)  ⇒ y^(−2/3)  = ((pr−mp^3 )/m)  ⇒my^(−2/3)  = pr−mp^3   ⇒ pr = my^(−2/3) +mp^3   ⇒ r = ((my^(−2/3) +mp^3 )/p)

$${y}\:=\:\left(\frac{{pr}}{{m}}−{p}^{\mathrm{3}} \right)^{−\mathrm{3}/\mathrm{2}} \\ $$$$\Rightarrow\:{y}^{−\mathrm{2}/\mathrm{3}} \:=\:\frac{{pr}}{{m}}−\frac{{mp}^{\mathrm{3}} }{{m}} \\ $$$$\Rightarrow\:{y}^{−\mathrm{2}/\mathrm{3}} \:=\:\frac{{pr}−{mp}^{\mathrm{3}} }{{m}} \\ $$$$\Rightarrow{my}^{−\mathrm{2}/\mathrm{3}} \:=\:{pr}−{mp}^{\mathrm{3}} \\ $$$$\Rightarrow\:{pr}\:=\:{my}^{−\mathrm{2}/\mathrm{3}} +{mp}^{\mathrm{3}} \\ $$$$\Rightarrow\:{r}\:=\:\frac{{my}^{−\mathrm{2}/\mathrm{3}} +{mp}^{\mathrm{3}} }{{p}} \\ $$

Commented by alephzero last updated on 17/Jun/22

y = −8, m = −1, p = 3  ⇒ r = ((−(−8)^(−2/3) −3^3 )/3) =  = −(((((−8)^2 ))^(1/3) −27)/3) = −((((64))^(1/3) −27)/3) =  = −((4−27)/3) = ((27−4)/3) = ((23)/3) = 7(2/3)

$${y}\:=\:−\mathrm{8},\:{m}\:=\:−\mathrm{1},\:{p}\:=\:\mathrm{3} \\ $$$$\Rightarrow\:{r}\:=\:\frac{−\left(−\mathrm{8}\right)^{−\mathrm{2}/\mathrm{3}} −\mathrm{3}^{\mathrm{3}} }{\mathrm{3}}\:= \\ $$$$=\:−\frac{\sqrt[{\mathrm{3}}]{\left(−\mathrm{8}\right)^{\mathrm{2}} }−\mathrm{27}}{\mathrm{3}}\:=\:−\frac{\sqrt[{\mathrm{3}}]{\mathrm{64}}−\mathrm{27}}{\mathrm{3}}\:= \\ $$$$=\:−\frac{\mathrm{4}−\mathrm{27}}{\mathrm{3}}\:=\:\frac{\mathrm{27}−\mathrm{4}}{\mathrm{3}}\:=\:\frac{\mathrm{23}}{\mathrm{3}}\:=\:\mathrm{7}\frac{\mathrm{2}}{\mathrm{3}} \\ $$

Answered by elviselloin719A last updated on 18/Jun/22

Ans: 7.667

$${Ans}:\:\mathrm{7}.\mathrm{667} \\ $$

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