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?

Question

Evelyn Arboleda · Accepted Answer

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#.

Jayden Jackson · Answer

undefined The volume of a rectangular prism is given by the formula $ V = l \times w \times h $, where $ l $ is the length, $ w $ is the width, and $ h $ is the height.

In this case, the height of the prism is given as $ x + 1 $, and the volume is given as $ x^3 + 7x^2 + 15x + 9 $. Since the height and width are equal, we can set $ w = x $.

Now, we can express the volume of the prism in terms of $ x $:

$$ V = l \times w \times h $$
$$ x^3 + 7x^2 + 15x + 9 = x \times x \times (x + 1) $$

Simplify the right side:

$$ x^3 + 7x^2 + 15x + 9 = x^3 + x^2 $$

Since the volume is equal, we have:

$$ x^3 + 7x^2 + 15x + 9 = x^3 + x^2 $$

Subtract $ x^3 $ from both sides:

$$ 7x^2 + 15x + 9 = x^2 $$

Subtract $ x^2 $ from both sides:

$$ 6x^2 + 15x + 9 = 0 $$

Now, we need to find the value of $ x $ that satisfies this equation. We can factorize it:

$$ 6x^2 + 15x + 9 = 3(2x^2 + 5x + 3) = 0 $$

$$ 2x^2 + 5x + 3 = 0 $$

Factorize further:

$$ 2x^2 + 2x + 3x + 3 = 0 $$

$$ 2x(x + 1) + 3(x + 1) = 0 $$

$$ (2x + 3)(x + 1) = 0 $$

Setting each factor to zero:

$$ 2x + 3 = 0 \Rightarrow x = -\frac{3}{2} $$

$$ x + 1 = 0 \Rightarrow x = -1 $$

Since the width cannot be negative, the width of the prism is $ x = -1 $.