Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. This concept extends the idea of direct variation to multiple variables and is often used in various scientific and engineering contexts.
joint variation
Joint variation occurs when a variable varies directly or inversely with multiple variables.
For instance:
If [latex]x[/latex] varies directly with both [latex]y[/latex] and [latex]z[/latex], we have [latex]x=kyz[/latex].
If [latex]x[/latex] varies directly with [latex]y[/latex] and inversely with [latex]z[/latex], we have [latex]x=\dfrac{ky}{z}[/latex].
Notice that we only use one constant in a joint variation equation.
A quantity [latex]x[/latex] varies directly with the square of [latex]y[/latex] and inversely with the cube root of [latex]z[/latex]. If [latex]x=6[/latex] when [latex]y=2[/latex] and [latex]z=8[/latex], find [latex]x[/latex] when [latex]y=1[/latex] and [latex]z=27[/latex].
Begin by writing an equation to show the relationship between the variables.
[latex]x=\dfrac{k{y}^{2}}{\sqrt[3]{z}}[/latex]
Substitute [latex]x=6[/latex], [latex]y=2[/latex], and [latex]z=8[/latex] to find the value of the constant [latex]k[/latex].
Now we can substitute the value of the constant into the equation for the relationship.
[latex]x=\dfrac{3{y}^{2}}{\sqrt[3]{z}}[/latex]
To find [latex]x[/latex] when [latex]y=1[/latex] and [latex]z=27[/latex], we will substitute values for [latex]y[/latex] and [latex]z[/latex] into our equation.