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 $144/(4xx4)=9$ .

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