A volume of a cube is given by V=L^3 where L is its side length. If a cube is dilated by a factor of 3.5, it means that its sidelength is increased by a factor of 3.5, i.e. if S is the first length, the new length L satisfies L=3.5*S. Now, the old cube's volume v was v=S^3, after it has expanded to the new sidelength L its new volume V is V=L^3. Using the equation L=3.5*S we can replace L in the equation for V like follows:
V=(3.5*S)^3
Expanding the product we get
V=[(3.5)^3]*(S^3)=42.875*(S^3)
We previously noticed that the prior volume of the cube was v=S^3. Replacing v for S^3 in the previeus equation gives us:
V=42.875v
Thus, the factor by wich the volume of the original cube was scaled up is 42.875