Joint Variation
\(\begin{aligned}x\propto\frac{x^m}{z^n}\end{aligned}\hspace{1mm}\text{(variation relation)}\) or
\(\begin{aligned} x=\frac{kx^m}{z^n} \end{aligned} \hspace{1mm} \text{(equation relation)}\)
such that
\(\begin{aligned} m&=1,2,3,\frac{1}{2},\frac{1}{3},\hspace{1mm} \\\\n&=1,2,3,\frac{1}{2},\frac{1}{3} \end{aligned}\)
and \(k\) is a constant.
\(\begin{aligned}\hspace{1mm}& y\propto \frac{x^2}{\sqrt{2}}\implies y = \frac{kx^2}{\sqrt{2}} \dots (1). \end{aligned}\)
Substitute \(y=8\), \( x=4\), and
\(z=36\) into \((1)\):
\(\begin{aligned}8&=\frac{k4^2}{\sqrt{36}}\implies k=\frac{(8)(6)}{16}\\\\&=3.\\\\ &\therefore y=\frac{3x^2}{\sqrt{z}}. \end{aligned}\)
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