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

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A rectangular prism's height is $x + 1$ $x+1$ . its volume is $x^{3} + 7 x^{2} + 15 x + 9$ $x^3+7x^2+15x+9$ . 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 $4$ $4$ units.

Explanation:

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

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

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

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

Dividing $x^{3} + 7 x^{2} + 15 x + 9$ $x^3+7x^2+15x+9$ by $(x^{2} + 2 x + 1)$ $(x^2+2x+1)$ ,

$x (x^{2} + 2 x + 1) + 5 (x^{2} + 2 x + 1) + 4 x + 4$ $x(x^2+2x+1)+5(x^2+2x+1)+4x+4$

But as volume is $l \times h \times w$ $lxxhxxw$ , $4 x + 4 = 4 (x + 1)$ $4x+4=4(x+1)$ too should be a multiple of $x^{2} + 2 x + 1 = {(x + 1)}^{2}$ $x^2+2x+1=(x+1)^2$ ,

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

Hence width is $4$ $4$ and height too is $4$ $4$

Note that volume is $3^{3} + 7 \times 3^{2} + 15 \times 3 + 9 = 27 + 63 + 45 + 9 = 144$ $3^3+7xx3^2+15xx3+9=27+63+45+9=144$

and length is $\frac{144}{4 \times 4} = 9$ $144/(4xx4)=9$ .

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

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Explanation:

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Explanation:

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