Question

The volume of a rectangular prism is represented by the function `x^3 + 11x^2 + 20x – 32` and The width of the box is x – 1 while the height is x + 8, how do you find the expression representing the length of the box?

Answer 1

Expression for the length of the box is `L=(x+4)`.

Explanation:

If `L`, `H` and `W` represent the length, height and width of the prism, then the volume of the rectangular prism is :

`V=L.H.W` ............. (1)

Given :
`V=x^3+11x^2+20x-32;` ............... (2)
`W=(x-1); \qquad H=(x+8)`.

Let `L=(x+l_0)` be the expression for the length, then the RHS of equation (1) becomes

`L.H.W = (x-l_0)(x+8)(x-1)`,
`\qquad \qquad \qquad \qquad = (x+l_0)(x^2+7x-8)`
`\qquad \qquad \qquad \qquad = (x+l_0)(x^2+7x-8)`
`\qquad \qquad \qquad \qquad = x^3+(7+l_0)x^2+(7l_0-8)x-8l_0` ..... (3)

Comparing this to the LHS of equation (1), we get the following set of equations to solve for `l_0`,
`7+l_0 = 11; \qquad 7l_0-8=20; \qquad 8l_0=32`;

`l_0=4`

Therefore `L=(x+4)`