Joint Variation


 Joint Variation

Definition joint variation
In general, for a combined variation\(y\) varies directly as \(x^m\) and inversely as \(z^n\) can be written as

\(\begin{aligned}x\propto\frac{x^m}{z^n}\end{aligned}\hspace{1mm}\text{(variation relation)}\) 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.

Example 3
Given that \(y\) varies directly as then square of \(x\)  varies inversely as square root of \(z\). If \(y=8\) when \(x=4\) and \(z=36\), express \(y\) in terms of \(x\) and \(z\).

\(\begin{aligned}\hspace{1mm}& y\propto \frac{x^2}{\sqrt{2}}\implies y = \frac{kx^2}{\sqrt{2}} \dots (1). \end{aligned}\)

Substitute \(y=8\)\( x=4\), and

\(z=36\) into \((1)\):

\(\begin{aligned}8&=\frac{k4^2}{\sqrt{36}}\implies k=\frac{(8)(6)}{16}\\\\&=3.\\\\ &\therefore y=\frac{3x^2}{\sqrt{z}}. \end{aligned}\)