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)?

Question

2417 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-12-10T18:10:26

#

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.#