lemme see, notice, this is the relationship from side ratios,
to areas and volumes
keep in mind that areas are square figures, involving 2 units,
and volumes are cubic figures, involving 3 units
thus [tex]\bf \begin{array}{llll}
&ratio&relationship\\
lenght&3:4&\cfrac{3}{4}=\cfrac{s}{s}\\\\
area&3:4&\left( \cfrac{3}{4} \right)^2=\cfrac{s^2(area)}{s^2(area)}\\\\
volume&3:4&\left( \cfrac{3}{4} \right)^3=\cfrac{s^3(volume)}{s^3(volume)}
\end{array}[/tex]
what the dickens all that means?
well, you have two cubes, both similar, their ratio, is 3:4,
3 is smaller than 4, thus is from smaller to bigger cube, 3:4 ratio
the smaller cube has a volume, or [tex]s^3[/tex] of 729 cubic units,
what's the other volume?
well, let us use those proportions above[tex]\bf \left( \cfrac{3}{4} \right)^3=\cfrac{729}{v}\implies \cfrac{3^3}{4^3}=\cfrac{729}{v}[/tex]
solve for "v"
notice, the numerator in 3:4, has the smaller volume, 729,
the bigger is at the bottom of that proportion
