SAT question about area and volume

[garveym2]

I am currently tutoring a student who is preparing for the SAT and I am hoping you can give me an explanation to a problem we are both stuck on:

The areas of the bottom, the side and the front of a rectangular box are r, s, and t square inches, respectively. What is the volume of the box, in cubic inches?

A. (rst)^3
B. (rst)^2
C. rst
D. square root of (rst)
E. third root of (rst)
[/i]

 

[skeeter]

let the dimensions of the box be L, W, and H

V = rH

V = sL

V = tW

V³ = (rH)(sL)(tW)

V³ = (rst)(LWH)

V³ = (rst)(V)

V² = rst

V = sqrt(rst)

 

[Denis]

V = LWH

LW = r [1]
LH = s [2]
WH = t [3]

[1] * [2] * [3]: L^2W^2H^2 = rst ; (LWH)^2 = rst ; LWH = sqrt(rst)

 

[Deleted member 4993]

Among all the answers only sqrt(rst) has the same dimension as the Vume - hence that must be the answer.

In SAT, sometimes you have to find answer like above to be fast!!!

 

[soroban]

Hello, garveym2!

The areas of the bottom, the side and the front of a rectangular box
are \(\displaystyle r,\:s,\:\) and \(\displaystyle t\) square inches, respectively.
What is the volume of the box, in cubic inches?

\(\displaystyle A.\; (rst)^3\;\;\;\;B.\; (rst)^2\;\;\;C.\; rst\;\;\;D.\;\sqrt{rst}\;\;\;E.\;\sqrt[3]{rst}\)

. . . - - * - - - - - *
. . . - -/- - - - - -/|
. . . - / - - - - - / |H
. . . -/- - - - - -/- |
. . . * - - - - - * - *
. . . | - - - - - | -/
. . .H| - - - - - | /W
. . . | - - - - - |/
. . . * - - - - - *
. . . - - - L

We know that: \(\displaystyle \:V \;=\;LWH\)

\(\displaystyle \begin{array}{ccccc}\text{The bottom has area }r: & \;LW & = & r & \;[1]\\
\text{The side has area }s: & \;WH & = & s & \;[2]\\
\text{The front has area }t: & \;LH & = & t & \;[3]\end{array}\)

Multiply [1], [2] and [3]: \(\displaystyle \LW)(WH)(LH) \:=\:rst\;\;\Rightarrow\;\;L^2W^2H^2\:=\:rst\)

Take the square root: \(\displaystyle \:LWH \:=\:\sqrt{rst}\)

Therefore: \(\displaystyle \:V \:=\:\sqrt{rst}\)

 

[garveym2]

Thank you!

Thank you all for your responses