What Are the Dimensions of the Food Bank's Shipping Container?

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The discussion revolves around solving for the dimensions of a rectangular shipping container used by the Food Bank, which has a volume of 2500 cm³. The container's dimensions are defined by the relationships: width is four times the depth, and height is five centimeters taller than the width. The cubic equation derived from these relationships is 16x³ + 20x² - 2500 = 0, where x represents the depth. Participants suggest using the Rational Root Theorem and factoring out the greatest common factor (G.C.F.) to find integer roots, as the equation does not yield simple factors.

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Homework Statement



A rectangular shipping container that the Food Bank uses to store their tinned food, has a volume of 2500 cm3. The container is 4 times as wide as it is deep, and 5cm taller than it is wide. What are the dimensions of the container


Homework Equations





The Attempt at a Solution



(x)(4x)(4x+5)

= 4x^3 + 20x^2 .

Usually when it asks for dimensions it wants the roots, However i went from factored to expanded, so wth am i supposed to do? can't use bionomial theorem b/c already know the facotred form, etc.

Help?
 
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You have an expression for the volume: (x)(4x)(4x+5 cm), where x is in cm. What this expression says is that if you specify the depth of the container, x, you can figure out the volume. You don't know the depth, though - what you know is the volume, so you have the reverse problem. The volume given is 2500 cm^3, so your task is to find the depth x such that

2500 cm^3 = (x)(4x)(4x+5 cm)

(Also, double-check your expanded form - you made a mistake).
 
yes, I understand what it is saying.. But what is my next step?

Or do i just combine the first two ? 16x^2 + 20x

Then use quadratic? or factor out? x(16x + 20 )

Idk..
 
It's a cubic equation. You can't use the quadratic formula. You have to solve the equation:

16x^3 + (20 cm)x^2 - 2500 cm^3 = 0

There's no nice-and-easy cubic formula that you can use to solve this. One way to solve it would be using a calculator or computer. However, perhaps you learned some tricks in class to try and solve cubic equations with integer coefficients? It turns out the relevant solution to the equation is rather simple.
 
Simplify the cubic equation into the form x^2 (...) = ..., I found it easier in that form.
 
verty said:
Simplify the cubic equation into the form x^2 (...) = ..., I found it easier in that form.
I don't see how this is helpful, since the right side won't be zero.
 
Ive tried solving it and i get 16x^3 + (20 cm)x^2 - 2500 cm^3 , but there is simply no factors that will make the eq = 0, so i can't use bionomial theorem. does it mean its not solveable without technology?
 
To make the coefficients a little bit more manageable, factor out the G.C.F. Then use the rational roots theorem. You'll find one integer root that is positive. The other two roots are complex.

And what is this bionomial theorem that you speak of? Never heard of it.
 

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