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Question-186337




Question Number 186337 by normans last updated on 03/Feb/23
Commented by normans last updated on 03/Feb/23
the volume of the cube is 300  the cube is cut with a plane.  the volume of the cut part????
$$\boldsymbol{{the}}\:\boldsymbol{{volume}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{cube}}\:\boldsymbol{{is}}\:\mathrm{300} \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{cube}}\:\boldsymbol{{is}}\:\boldsymbol{{cut}}\:\boldsymbol{{with}}\:\boldsymbol{{a}}\:\boldsymbol{{plane}}. \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{volume}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{cut}}\:\boldsymbol{{part}}???? \\ $$
Answered by mr W last updated on 03/Feb/23
Commented by mr W last updated on 03/Feb/23
a=edge length of cube  volume of cube V=a^3 =300  (b/a)=((0.4a)/(0.5a))  ⇒b=0.8a  (h/(h+a))=((0.4a)/(0.5a))=0.8  ⇒h=4a  V_(cut) =V_1 +V_2   V_1 =((0.4a×0.8a)/2)×(a/3)=((4a^3 )/(75))  V_2 =(((0.4a+0.5a)×a)/2)×(a/3)=((3a^3 )/(20))  V_(cut) =((4a^3 )/(75))+((3a^3 )/(20))=((61a^3 )/(300))=((61V)/(300))=61 ✓  or  V_(cut) =((0.5a×a)/2)×((4a+a)/3)−((0.4a×0.8a)/2)×((4a)/3)         =((61a^3 )/(300))=((61V)/(300))=61 ✓
$${a}={edge}\:{length}\:{of}\:{cube} \\ $$$${volume}\:{of}\:{cube}\:{V}={a}^{\mathrm{3}} =\mathrm{300} \\ $$$$\frac{{b}}{{a}}=\frac{\mathrm{0}.\mathrm{4}{a}}{\mathrm{0}.\mathrm{5}{a}} \\ $$$$\Rightarrow{b}=\mathrm{0}.\mathrm{8}{a} \\ $$$$\frac{{h}}{{h}+{a}}=\frac{\mathrm{0}.\mathrm{4}{a}}{\mathrm{0}.\mathrm{5}{a}}=\mathrm{0}.\mathrm{8} \\ $$$$\Rightarrow{h}=\mathrm{4}{a} \\ $$$${V}_{{cut}} ={V}_{\mathrm{1}} +{V}_{\mathrm{2}} \\ $$$${V}_{\mathrm{1}} =\frac{\mathrm{0}.\mathrm{4}{a}×\mathrm{0}.\mathrm{8}{a}}{\mathrm{2}}×\frac{{a}}{\mathrm{3}}=\frac{\mathrm{4}{a}^{\mathrm{3}} }{\mathrm{75}} \\ $$$${V}_{\mathrm{2}} =\frac{\left(\mathrm{0}.\mathrm{4}{a}+\mathrm{0}.\mathrm{5}{a}\right)×{a}}{\mathrm{2}}×\frac{{a}}{\mathrm{3}}=\frac{\mathrm{3}{a}^{\mathrm{3}} }{\mathrm{20}} \\ $$$${V}_{{cut}} =\frac{\mathrm{4}{a}^{\mathrm{3}} }{\mathrm{75}}+\frac{\mathrm{3}{a}^{\mathrm{3}} }{\mathrm{20}}=\frac{\mathrm{61}{a}^{\mathrm{3}} }{\mathrm{300}}=\frac{\mathrm{61}{V}}{\mathrm{300}}=\mathrm{61}\:\checkmark \\ $$$${or} \\ $$$${V}_{{cut}} =\frac{\mathrm{0}.\mathrm{5}{a}×{a}}{\mathrm{2}}×\frac{\mathrm{4}{a}+{a}}{\mathrm{3}}−\frac{\mathrm{0}.\mathrm{4}{a}×\mathrm{0}.\mathrm{8}{a}}{\mathrm{2}}×\frac{\mathrm{4}{a}}{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:=\frac{\mathrm{61}{a}^{\mathrm{3}} }{\mathrm{300}}=\frac{\mathrm{61}{V}}{\mathrm{300}}=\mathrm{61}\:\checkmark \\ $$
Commented by normans last updated on 03/Feb/23
nice Sir
$$\boldsymbol{{nice}}\:\boldsymbol{{Sir}} \\ $$

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