The width of a plastic storage box is 2 ft longer than the height. The length is 6 ft longer than the height. The volume is 240 ft3. Answer the following questions.

Question

What is the formula for the volume of a rectangular​ prism?

What variable expressions represent the​ length, height, and​ width?

What equation represents the volume of the plastic storage​ box?

What are the dimensions of the​ box?

Answer 1

V = LWH

volume = length x width x height

240 = (2+H)((6+H)H = (H^2 + 8H + 12)H = H^3 + 8H^2 + 12H

H^3 + 8H^2 + 12H - 240 = 0

while there's no easy way to solve cubics, try some simple integers. You need an even integer

2 doesn't work, too small, try 4

4^3 + 8(4)^2 + 12(4) -240 =0 Height = 4 feet

dimensions are 4 x 10 x 6 feet = 240 ft/63

divide h-4 into the cubic to get H^2 + 12H + 60

the discriminate = 144-4(60) < 0 so there are no other real solutions

4 feet by 6 feet by 10 feet is the only solution

Answer 2

We will let h be the height of the box.

Given: l = h+6

Given: w = h+2

Given: V=240ft^3

V=(l)(w)(h) so

240=(h+6)(h+2)(h)

240=h^3+8h^2+12h

0=h^3+8h^2+12h-240

I look at the coefficients and notice that all of the non leads are divisible by 4, so I am going to see if it is a solution of the problem by using synthetic division.

4| 1 8  12 -240

   __4 48 240

   1 12 60 | 0

Since the remainder is 0, 4 is a root of the equation

That means we can factor to get: (h-4)(h^2+12h+60)=0

The other two roots of the equation are imaginary so,

H=4, L=10, W=6 by substitution.