A cubic shipping container had a volume of v^3 cubic meters. The height of the
container was decreased by a whole number of meters and the width was increased
by a whole number of meters so that the volume of the container is now 
v^3 + 2v^2 – 3v. Find out the following:

a.	By how many meters the height was decreased?

b.	By how many meters the width was increased?  

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volume of container originally = v³ = 
 original length × original width × original height

Let l = original length
Let w = original width
Let h = original height

Let a = the whole number of meters the height was decreased

Let b = the whole number of meters the width was increased

Then

new length = original length = l (length was not changed)

new width = (original width + b) = w + b

new height = (original height - a) = h - a

new volume = v³ + 2v² - 3v  

new volume = new length × new width × new height

v³ + 2v² - 3v = l(w + b)(h - a)

v³ + 2v² - 3v = l(w + b)(h - a)

v³ + 2v² - 3v = l(wh - aw + bh - ab)

v³ + 2v² - 3v = lwh - law + lbh - lab

Since v³ = lwh we subtract v³ from left side and
subtract lwh from right side


     2v² - 3v = -law + lbh - lab

Rearranging terms on righ so that positive term comes first

     2v² - 3v = lbh - law - lab 

Factor out v on left; factor out l on right

    v(2v - 3) = l(bh - aw - ab)

As it turns out there are infinitely many solutions to that.

Here are a few:

	v = 2, l = 1, w = 1, b = 1, a = 3, h = 8

original volume = original length × original width × original height
original volume = lwh = 1(1)(8) = 8
original volume = v³ = 2³ = 8 
That checks!
new length = original length = 1 (length was not changed)
new width = (original width + b) = 1 + 1 = 2
new height = (original height - a) = 8 - 3 = 5
new volume = v³ + 2v² - 3v = 2³ + 2(2)² - 3(2) = 10  
new volume = new length × new width × new height
new volume = 1(2)(5) = 10
That checks!  Answer: height decreased by a = 3,
width increased by b = 1
=========================================
        v = 5, l = 5, w = 5, b = 11, a = 3, h = 5

original volume = original length × original width × original height
original volume = lwh = 5(5)(5) = 125
original volume = v³ = 5³ = 125
That checks! 
new length = original length = 5 (length was not changed)
new width = (original width + b) = 5 + 11 = 16
new height = (original height - a) = 5 - 3 = 2
new volume = v³ + 2v² - 3v = 5³ + 2(5)² - 3(5) = 160  
new volume = new length × new width × new height
new volume = 5(16)(2) = 160
That checks!  Answer: height decreased by a = 3,
width increased by b = 11
=========================================
        v = 6, l = 9, w = 4, b = 6, a = 3, h = 6

original volume = original length × original width × original height
original volume = lwh = 9(4)(6) = 216
original volume = v³ = 6³ = 216
That checks!
new length = original length = 9 (length was not changed)
new width = (original width + b) = 4 + 6 = 10
new height = (original height - a) = 6 - 3 = 3
new volume = v³ + 2v² - 3v = 6³ + 2(6)² - 3(6) = 270  
new volume = new length × new width × new height
new volume = 9(10)(3) = 270
That checks!  Answer: height decreased by a = 3,
width increased by b = 6
=========================================
        v = 7, l = 7, w = 7, b = 5, a = 2, h = 7

original volume = original length × original width × original height
original volume = lwh = 7(7)(7) = 343
original volume = v³ = 7³ = 343
That checks!
new length = original length = 7 (length was not changed)
new width = (original width + b) = 7 + 5 = 12
new height = (original height - a) = 7 - 2 = 5
new volume = v³ + 2v² - 3v = 7³ + 2(7)² - 3(7) = 420  
new volume = new length × new width × new height
new volume = 7(12)(5) = 420
That checks!  Answer: height decreased by a = 2,
width increased by b = 5
=========================================
        v = 12, l = 12, w = 12, b = 3, a = 1, h = 12

original volume = original length × original width × original height
original volume = lwh = 12(12)(12) = 1728
original volume = v³ = 12³ = 1728
That checks!
new length = original length = 12 (length was not changed)
new width = (original width + b) = 12 + 3 = 15
new height = (original height - a) = 12 - 1 = 11
new volume = v³ + 2v² - 3v = 12³ + 2(12)² - 3(12) = 1980  
new volume = new length × new width × new height
new volume = 12(15)(11) = 1980
That checks!  Answer: height decreased by a = 1,
width increased by b = 3
=========================================
        v = 15, l = 15, w = 15, b = 13, a = 6, h = 15

original volume = original length × original width × original height
original volume = lwh = 15(15)(15) = 3375
original volume = v³ = 15³ = 3375
That checks!
new length = original length = 15 (length was not changed)
new width = (original width + b) = 15 + 13 = 28
new height = (original height - a) = 15 - 6 = 9
new volume = v³ + 2v² - 3v = 15³ + 2(15)² - 3(15) = 3780  
new volume = new length × new width × new height
new volume = 15(28)(9) = 3780
That checks!  Answer: height decreased by a = 6,
width increased by b = 13
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There are billions and billions more solutions!!! 

Edwin