# Given volume and SA, find lengths.

• Dec 26th 2007, 09:15 AM
amardeep33
[Solved!]Given volume and SA, find lengths.
Given volume and surface area of a rectangular solid, and sides are in GP.?
The volume of a certain recangular solid is 64 and its total surface area is 384. Its three dimensions are in geometric progression. What is the sum of the lengths of all the edges of this solid.

I myself don't really understand what the last sentence means, but I believe its something like finding the perimeter, but of a 3D object. Anyway, since the sides are in a GP, I know that the sides are x, xr, and xr². Using the volume formula, you can simplify to get
$x = 4/r$.
I'm having trouble with the surface area part, as when I substitute this x value into the forumla
( $2x^2r + 2x^2r^2 + 2x^2r^3$), I get a quadratic which has two different values for r, but I am pretty sure there is only supposed to be one value for r, otherwise there would be MANY solutions to this question. Thanks in advance for any help.
• Dec 26th 2007, 09:51 AM
Isomorphism
Are you sure??
I got a quadratic like r^2 - 11r + 1 = 0. In the two solutions that follow,one of them is negative. We keep the positive solution but discard the negative because......

We claim this is not possible because if r < 0 then x < 0 (xr = 4, remember?)
So xr^2 < 0. Contradiction because length of an edge cannot be negative!
Q.E.D:D
• Dec 26th 2007, 10:08 AM
amardeep33
I had got that at first, which is what made me post this.
Since then, I've asked someone who has helped me greatly.
I'll quote him directly incase you actually want to figure out this problem. The main difference is leaving in an r after substitution to make it a little easier on the eyes.
Here it is.
Quote:

let x be the width of the shortest side
the other sides has xr, and xr^2 in length
volume= $x^3r^3=64$
xr=4
area= $384=2(x^2r+x^2r^2+x^2r^3)$
substitube into the equ
and get 192 = 4x + 16 +16r
176=4x+16r
44=x+4r
44=x+4(4/x)
44x=x^2+16
x^2-44x+16=0
x=44+-sqrt1872 all over 2
x=43.6 or 0.36
the correct length is 43.6, then r=0.09
the dimensions are 43.6, 4, 0.36
• Dec 26th 2007, 10:48 AM
Isomorphism
Quote:

Originally Posted by amardeep33
I had got that at first, which is what made me post this.
Since then, I've asked someone who has helped me greatly.
I'll quote him directly incase you actually want to figure out this problem. The main difference is leaving in an r after substitution to make it a little easier on the eyes.
Here it is.

That's what I said too. In fact you(or that helper) substituted for r and I did it for x.
At this step 44=x+4r in the proof, substitute x = 4/r and you will get $r^2 - 11r + +1 = 0$, then you can follow my reasoning. (>looks< like it is another solution no??).

On a side note "x=43.6 or 0.36"--- Shouldn't this read "x=43.6 or -0.36"??
• Dec 26th 2007, 11:40 AM
amardeep33
Ah. Yes, i guess the -11 way works also, but when I did it at first I was confused by the fact that there were two solutions.
And no, its not negative .3667, because if it was, then you would have a negative side (xr).
You might have a calculation error.
Thanks for the help.
• Dec 26th 2007, 11:59 AM
Isomorphism
Quote:

Originally Posted by amardeep33
Ah. Yes, i guess the -11 way works also, but when I did it at first I was confused by the fact that there were two solutions.
And no, its not negative .3667, because if it was, then you would have a negative side (xr).
You might have a calculation error.
Thanks for the help.

Ya you are right :D
I had a computation error, but why do you say "the correct length is 43.6, then r=0.09"?? What is the reason you discard the other solution? You claim let x be the shortest length, I say that itself is eliminating your other solution.

What do you have against x=+0.3667??
• Dec 26th 2007, 12:53 PM
amardeep33
Lol. What do I have against .3667?
Well, first off, its one hell of an ugly number.
But it doesnt matter which of the 2 solutions you use, as my goal was to find the 3 sides and add them, and if you multiply $43.6 * .09^2$(which represents the side $xr^2$ , you get .3667. The same thing happens if you use .3667 as x instead of 43.6.
• Dec 26th 2007, 01:16 PM
Isomorphism
Quote:

But it doesnt matter which of the 2 solutions you use, as my goal was to find the 3 sides and add them, and if you multiply 43.6 * .09^2(which represents the side xr^2 , you get .3667. The same thing happens if you use .3667 as x instead of 43.6.
Yes! I know :rolleyes:
Generally quadratics behave that way :)
But you should say that in your solution. That is what I was trying to point out. You should not merely claim "x=46 is the required length.....". But since you know the reason, I have no problem at all :D

Do well, may god bless you :)

P.S: Don't hate decimal representations like that. They save your butt henceforth :D