Question 193523
construct a function expressing the volume V of a cube as as function of the 
length of its diagonal d, where d is the line segment joining opposite
 non- co- planer vertices of the cube and d is not a side of the cube.
:
drawing this will help:
The hypotenuse they are talking about is formed by the diagonal of the bottom of the cube and the height of the cube
:
Let x = side of the cube
then
{{{sqrt(2x^2)}}} = diagonal of the bottom
and
d = {{{sqrt((sqrt(2x^2))^2 + x^2)}}}
Which is
d = {{{sqrt(2x^2 + x^2)}}}
d = {{{sqrt(3x^2)}}}
Solve for x
d^2 = 3x^2
{{{d^2/3}}} = x^2
which is
x = {{{sqrt(d^2/3)}}}
:
v = x^3
Substitute {{{sqrt(d^2/3)}}} for x:
V(d) = {{{(sqrt(d^2/3))^3}}}; is the required function
:
I checked this using x=2, you can do the same