How do you Find the volume of the solid that lies in the first octant and is bounded by the three coordinate planes and another plane passing through (3,0,0), (0,4,0), and (0,0,5)?

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Answer 1 · 2017-12-10T18:10:26

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The solid in question is a tetrahedron #V-ABC# having vertices

#O(0,0,0), A(3,0,0), B(0,4,0) and C(0,0,5).#

Its volume #V# is given by,

#V=|1/6[vec(OA),vec(OB),vec(OC)]|,#

where, #[vec(VA),vec(VB),vec(VC)]# denotes the scalar triple product

#vec(VA)*{vec(VB)xxvec(VC)}.#

We have, #vec(VA)=A(3,0,0)-O(0,0,0)=(3,0,0).#

Similarly, #vec(OB)=(0,4,0), and vec(OC)=(0,0,5).#

#:.[vec(VA),vec(VB),vec(VC)]=|(3,0,0),(0,4,0),(0,0,5)|,#

#=3*4*5=60.#

#rArr V=1/6|60|=10.#