A rectangular prism's height is #x+1#. its volume is #x^3+7x^2+15x+9#. If height and width of the prism are equal, what is its width?

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Answer 1 · 2017-01-03T06:45:31

Width of the prism is #4# units.

As the volume of a rectangular prism, whose length is #l#, height is #h# and width is #w# is #lxxhxxw#.

As the volume of rectangular prism is #x^3+7x^2+15x+9#,

and height is #(x+1)# and width and height being same, height too is #(x+1)#

we can have its length by dividing #x^3+7x^2+15x+9# by #(x+1)(x_1)=x^2+2x+1#.

Dividing #x^3+7x^2+15x+9# by #(x^2+2x+1)#,

#x(x^2+2x+1)+5(x^2+2x+1)+4x+4#

But as volume is #lxxhxxw#, #4x+4=4(x+1)# too should be a multiple of #x^2+2x+1=(x+1)^2#,

which is possible if #x+1=4# i.e. #x=3#

Hence width is #4# and height too is #4#

Note that volume is #3^3+7xx3^2+15xx3+9=27+63+45+9=144#

and length is #144/(4xx4)=9#.