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?

1 Answer
Feb 28, 2016

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)