Hi Gina,

When the statement is that "the scale is 0.75" it means that the linear scale is 0.75. Since $0.75 = \large \frac34$ the statement that "the scale is 0.75" means that

\[\mbox{(a linear measurement of the smaller prism)}= \frac34 \times \mbox{(the same linear measurement on the larger prism.)}\]

Multiplying both sides of this equation by $\large \frac43$ gives

\[\mbox{(a linear measurement of the larger prism)}= \frac43 \times \mbox{(the same linear measurement on the smaller prism.)}\]

Suppose what you have is a right rectangular prism so its area is

\[\mbox{length} \times\mbox{ width} \times \mbox{ height.}\]

Suppose the length, width and height of the larger prism are $L, W$ and $H$ and for the smaller prism are $l, w$ and $h.$ Then

\[L = \frac43 l, W = \frac43 w \mbox{ and } H = \frac43 h\]

and thus

\[L \times W \times H = \frac43 l \times \frac43 w \times \frac43 h = \left(\frac43 \right)^3 l \times w \times h.\]

Thus the volume of the larger prism is $\large \left(\frac43 \right)^3$ times the volume of the smaller prism.

The factor of the cube of the scaling factor applies in many situations. If you scale up a three dimensional object by a linear scale of $k$ then the volume of the scaled up object is $k^3$ times the volume of the smaller object.

Penny