# Cubes, Conic Sections, and Crockett Johnson - Conic Sections and Doubling a Cube

Author(s):
Stephanie Cawthorne (Trevecca Nazarene University) and Judy Green (Marymount University) Figure 7. Two parabolas meeting a hyperbola at the point $$(s,t).$$ (Constructed using Geometer’s Sketchpad.)

Given a cube with edge of length $$a,$$ how do we now utilize conic sections to double that cube; i.e., how do we find $$x$$ so that $$x^3=2a^3$$?  Consider the point of intersection, $$(s,t),$$ in Figure 7 above.  The curves in this figure represent the parabolas $$x^2=ay$$ and $$y^2=2ax,$$ together with the hyperbola $$xy=2a^2;$$ i.e., the graphs of the functions $y={\frac{x^2}{a}}, {y=\sqrt{2ax}},\,\,{\rm{and}}\,\,{y={\frac{2a^2}{x}}},$ for $$x>0.$$  The equation of the first function tells us that $$t={{s^2}/{a}},$$ so that the equation of the second parabola becomes ${\left({\frac{s^2}{a}}\right)}^2={2as},\,\,{\rm or}\,\,\,s^3=2a^3.$ Thus the $$x$$-coordinate of the point of intersection, $$s,$$ is the length of the edge of a cube which is double the volume of the original cube with edge of length $$a.$$  If we look at the cube of the $$y$$-coordinate of the point we find $t^3={\frac{s^6}{a^3}}={\frac{(s^3)^2}{a^3}}={\frac{4a^6}{ a^3}}=4a^3$ so the $$y$$-coordinate is the length of the edge of a cube which is quadruple the volume of the original cube, or the larger of the mean proportionals mentioned previously.