Question Number 52470 by ajfour last updated on 08/Jan/19
Commented by ajfour last updated on 08/Jan/19
$${Find}\:{r}\:{in}\:{terms}\:{of}\:{a},{b},{c}. \\ $$$$\left({OD}\:{is}\:{vertical}\right). \\ $$
Answered by mr W last updated on 08/Jan/19
$${V}=\frac{\mathrm{1}}{\mathrm{3}}×{abc}\:={pyramid}\:{ACBO}−{D}\:\left({OD}\right) \\ $$$${V}_{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{bc}}{\mathrm{2}}\right){r}\:=\:{pyramid}\:{AOD}−{M}\:\left({TM}\right) \\ $$$${V}_{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{ac}}{\mathrm{2}}\right){r}\:={pyramid}\:{COD}−{M}\:\left({SM}\right) \\ $$$${V}_{\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{a}\sqrt{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }}{\mathrm{2}}\right){r}={pyramid}\:{ABD}−{M}\:\left({QM}\right) \\ $$$${V}_{\mathrm{4}} =\frac{\mathrm{1}}{\mathrm{3}}\left(\frac{{b}\sqrt{{a}^{\mathrm{2}} +{c}^{\mathrm{2}} }}{\mathrm{2}}\right){r}={pyramid}\:{CBD}−{M}\:\left({PM}\right) \\ $$$${V}={V}_{\mathrm{1}} +{V}_{\mathrm{2}} +{V}_{\mathrm{3}} +{V}_{\mathrm{4}} \\ $$$$\Rightarrow{r}=\frac{\mathrm{2}{abc}}{\left({a}+{b}\right){c}+{a}\sqrt{{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }+{b}\sqrt{{a}^{\mathrm{2}} +{c}^{\mathrm{2}} }} \\ $$
Commented by ajfour last updated on 08/Jan/19
$${Thank}\:{you}\:{Sir},\:{too}\:\mathcal{M}{arvelous}! \\ $$
Commented by mr W last updated on 08/Jan/19
