Example  4

Yvonne built a model of a cube-shaped cabin she plans to build on her acreage. The model has a volume of \(15 \thinspace 625 \thinspace \rm{cm}^3\).

  1. Determine the side length of her model.

    Because all the side lengths are equal in a cube, \(V = s^3\).

    To find the side length, take the cube root of both sides.

    \(\begin{align}
     s_1 &= \sqrt[3]{V} \\ 
     s_1 &= \sqrt[3]{{15\thinspace 625}} \\ 
     s_1 &= 25 \\ 
     \end{align}\)

    Each side is \(25\) cm long.

  2. Yvonne finds the model too small to work with, and wants to double the volume of the cube. By what scale factor does she have to increase the side length to accomplish this?

    Return to the equation for side length, and change the volume to \(2V\).

    \(\begin{align}
     s_2 &= \sqrt[3]{V} \\ 
     s_2 &= \sqrt[3]{{2V}} \\ 
     \end{align}\)

    To determine the scale factor, divide the new side length by the old side length.

    \(\begin{align}
     \frac{{s_2 }}{{s_1 }} &= \frac{{\sqrt[3]{{2V}}}}{{\sqrt[3]{V}}} \\ 
      &= \sqrt[3]{2} \\ 
      &= 1.259... \\ 
       &\doteq 1.26 \\ 
     \end{align}\)

    Yvonne must use a scale factor of approximately \(1.26\) in order to double the volume of the model.