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Question Number 20149 by ajfour last updated on 22/Aug/17

Commented by ajfour last updated on 22/Aug/17

The roof is even a flat plane.Lengths a, b, c, y are perpendicular to the bottom rectangular face with sides h and l . Find height y and the volume enclosed by the surfaces.

Theroofisevenaflatplane.Lengthsa,b,c,yareperpendiculartothebottomrectangularfacewithsideshandl.Findheightyandthevolumeenclosedbythesurfaces.

Answered by mrW1 last updated on 22/Aug/17

((a+y)/2)=((b+c)/2) ⇒y=b+c−a V=l×h×((a+b+c+y)/4)=l×h×((a+b+c+b+c−a)/4) =((lh(b+c))/2)

a+y2=b+c2⇒y=b+c−aV=l×h×a+b+c+y4=l×h×a+b+c+b+c−a4=lh(b+c)2

Commented by ajfour last updated on 22/Aug/17

thank you Sir, you make it so simple!

thankyouSir,youmakeitsosimple!

Answered by ajfour last updated on 23/Aug/17

let eqn. of plane be (r^� −a^� ).n^� =0 let origin be the bottom left corner. n^� =[li^� +(b−a)k^�^� ]×[hj^� +(c−a)k^� ] =−h(b−a)i^� −l(c−a)j^� +lhk^� ] ⇒ −h(b−a)x−l(c−a)y+lhz=alh z=a+(((b−a)x)/l)+(((c−a)y)/h) at x=l, y=h , z=y (asked) so y=a+b−a+c−a y+a=b+c ⇒ y=b+c−a V=∫_0 ^( l) (∫_0 ^( h) zdy)dx =∫_0 ^( l) [ah+(((b−a)hx)/l)+(((c−a)h)/2)]dx =ahl+(((b−a)lh)/2)+(((c−a)lh)/2) =((lh)/2)(2a+b−a+c−a) Volume V=((lh(b+c))/2) .

leteqn.ofplanebe(r¯−a¯).n^=0letoriginbethebottomleftcorner.n¯=[li^+(b−a)k^]×[hj^+(c−a)k^]=−h(b−a)i^−l(c−a)j^+lhk^]⇒−h(b−a)x−l(c−a)y+lhz=alhz=a+(b−a)xl+(c−a)yhatx=l,y=h,z=y(asked)soy=a+b−a+c−ay+a=b+c⇒y=b+c−aV=∫0l(∫0hzdy)dx=∫0l[ah+(b−a)hxl+(c−a)h2]dx=ahl+(b−a)lh2+(c−a)lh2=lh2(2a+b−a+c−a)VolumeV=lh(b+c)2.

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