# finding volume with algebra

• Dec 17th 2006, 11:15 AM
puppy_wish
finding volume with algebra

PUZZLING PRISM:
The areas of three faces of a rectangular prism are 63 cm2, 56 cm2 and 72 cm2. What is the volume of the prism?

What I know:

Volume= LxWxH
Area = LxW

• Dec 17th 2006, 11:24 AM
ThePerfectHacker
Quote:

Originally Posted by puppy_wish

PUZZLING PRISM:
The areas of three faces of a rectangular prism are 63 cm2, 56 cm2 and 72 cm2. What is the volume of the prism?

What I know:

Volume= LxWxH
Area = LxW

You know that,
$\displaystyle xy=63$
$\displaystyle xz=56$
$\displaystyle yz=72$
Multiply them together,
$\displaystyle xyxzyz=63\cdot 56\cdot 72=254016$
$\displaystyle x^2y^2z^2=(xyz)^2=V^2=254016$
Thus,
$\displaystyle V=\sqrt{254016}=504$
• Dec 17th 2006, 11:33 AM
Soroban
Hello, puppy_wish!

This can be solved with "normal" algebra
. . but there is a cute solution . . .

Quote:

The areas of three faces of a rectangular prism are 63 cm², 56 cm² and 72 cm².
What is the volume of the prism?

Let the length, width, height be: $\displaystyle L,\,W,\,H.$

We are told that: .$\displaystyle \begin{array}{ccc} LW \,=\,63 \\ WH\,=\,56 \\ LW\,=\,72\end{array}$

Multiply the three equations: .$\displaystyle (LW)(WH)(LH) \:=\:(63)(56)(72)$

and we have: .$\displaystyle L^2W^2H^2 \:=\:254,016\quad\Rightarrow\quad(LWH)^2\:=\:254,0 16$

Therefore: .$\displaystyle LWH \:=\:\sqrt{254,016} \:=\:504\text{ cm}^2$

I like this solution . . . I hope you do.
We answered the questi0on without finding $\displaystyle L,\,W$ and $\displaystyle H.$

• Dec 17th 2006, 11:39 AM
puppy_wish
Quote:

Originally Posted by Soroban
Hello, puppy_wish!

This can be solved with "normal" algebra
. . but there is a cute solution . . .

Let the length, width, height be: $\displaystyle L,\,W,\,H.$

We are told that: .$\displaystyle \begin{array}{ccc} LW \,=\,63 \\ WH\,=\,56 \\ LW\,=\,72\end{array}$

Multiply the three equations: .$\displaystyle (LW)(WH)(LH) \:=\:(63)(56)(72)$

and we have: .$\displaystyle L^2W^2H^2 \:=\:254,016\quad\Rightarrow\quad(LWH)^2\:=\:254,0 16$

Therefore: .$\displaystyle LWH \:=\:\sqrt{254,016} \:=\:504\text{ cm}^2$

I like this solution . . . I hope you do.
We answered the questi0on without finding $\displaystyle L,\,W$ and $\displaystyle H.$

wow thank you both of you. I really appreciate it :D