Question-52470

Question Number 52470 by ajfour last updated on 08/Jan/19

Commented by ajfour last updated on 08/Jan/19

Find r in terms of a,b,c. (OD is vertical).

$${Find}\:{r}\:{in}\:{terms}\:{of}\:{a},{b},{c}. \\ $$$$\left({OD}\:{is}\:{vertical}\right). \\ $$

Answered by mr W last updated on 08/Jan/19

V=(1/3)×abc =pyramid ACBO−D (OD) V_1 =(1/3)(((bc)/2))r = pyramid AOD−M (TM) V_2 =(1/3)(((ac)/2))r =pyramid COD−M (SM) V_3 =(1/3)(((a(√(b^2 +c^2 )))/2))r=pyramid ABD−M (QM) V_4 =(1/3)(((b(√(a^2 +c^2 )))/2))r=pyramid CBD−M (PM) V=V_1 +V_2 +V_3 +V_4 ⇒r=((2abc)/((a+b)c+a(√(b^2 +c^2 ))+b(√(a^2 +c^2 ))))

$${V}=\frac{\mathrm{1}}{\mathrm{3}}×{abc}\:={pyramid}\:{ACBO}−{D}\:\left({OD}\right) \\ $$$${V}_{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{bc}}{\mathrm{2}}\right){r}\:=\:{pyramid}\:{AOD}−{M}\:\left({TM}\right) \\ $$$${V}_{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{ac}}{\mathrm{2}}\right){r}\:={pyramid}\:{COD}−{M}\:\left({SM}\right) \\ $$$${V}_{\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{a}\sqrt{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }}{\mathrm{2}}\right){r}={pyramid}\:{ABD}−{M}\:\left({QM}\right) \\ $$$${V}_{\mathrm{4}} =\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{b}\sqrt{{a}^{\mathrm{2}} +{c}^{\mathrm{2}} }}{\mathrm{2}}\right){r}={pyramid}\:{CBD}−{M}\:\left({PM}\right) \\ $$$${V}={V}_{\mathrm{1}} +{V}_{\mathrm{2}} +{V}_{\mathrm{3}} +{V}_{\mathrm{4}} \\ $$$$\Rightarrow{r}=\frac{\mathrm{2}{abc}}{\left({a}+{b}\right){c}+{a}\sqrt{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }+{b}\sqrt{{a}^{\mathrm{2}} +{c}^{\mathrm{2}} }} \\ $$

Commented by ajfour last updated on 08/Jan/19