Question

The areas of three faces of a right rectangular prism are 5, 7, and 11. What is its volume?

Answer 1

>Take the base to be the triangle of side 5, 7,11

If the solid is a right, rectangular prism, then its bases will be rectangles, not a triangles.

Answer 2

Let x and y be the sides of the base and z the height. We have
\[ x y = 5
x z = 7
y z = 11

]
We need V = xyz.
Can you manage it from here?

Answer 3

To find x and y, notice that by dividing the last two equations
\[ \frac x y =\frac 7 {11}\\
x = \frac {7 y} {11}

\]
replace in the first equation, you get
\[
x y = 5\\
\frac {7 y} {11} y= 5 \\
7 y^2 = 55\\
y^2 = \frac {55}{11}\\
y= \frac { \sqrt {55} }{\sqrt{11}}

\]
Find x, then z .
Your answer should be x y z

Answer 4

\[
x = \frac 7 {11} y=\frac 7 {11} \frac { \sqrt {55} }{\sqrt{11}}=\frac { \sqrt {35} }{\sqrt{11}}
\]

Answer 5

\[
x = \frac 7 {11} \\y=\frac 7 {11} \frac { \sqrt {55} }{\sqrt{11}}\\
z= \frac {11}y= 11 \frac { \sqrt {11} }{\sqrt{55}}=\frac { \sqrt {77} }{\sqrt{5}}\\\\
\]

Answer 6

Sqrt[35/11] Sqrt[55/7] Sqrt[77/5]
Now compute
\[
V = x y z=\sqrt { \frac {(35)(55)(77)}
{(11)(7) (5)}
}=\sqrt{385}
\]

Answer 7

A quickest way is to multiply the three original equations
\[
x^2 y^2 z^2 = (5)(7)(11)= 385\\
x y z =\sqrt{385}
\]