[tex]\qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad \stackrel{\textit{constant of variation}}{y=\stackrel{\downarrow }{k}x~\hfill } \\\\ \textit{\underline{x} varies directly with }\underline{z^5}\qquad \qquad \stackrel{\textit{constant of variation}}{x=\stackrel{\downarrow }{k}z^5~\hfill } \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{ \begin{array}{llll} \textit{\tiny "y" varies directly}\\ \textit{\tiny with the square of "x" } \end{array}}{y = kx^2}\qquad \textit{we also know that} \begin{cases} x = 8\\ y = 32 \end{cases} \\\\\\ 32=k(8)^2\implies \cfrac{32}{8^2}=k\implies \cfrac{1}{2}=k~\hfill \boxed{y=\cfrac{1}{2}x^2} \\\\\\ \textit{when y = 112.5, what is "x"?}\qquad 112.5=\cfrac{1}{2}x^2 \\\\\\ 225=x^2\implies \sqrt{225}=x\implies 15=x[/tex]
