Question

If $y$ varies indirectly as the cube of $x$, and $k$ is the constant variation, how do you find the equation to describe this relationship?

Answer 1

Hint:The statement: $y$ indirectly as the cube of $x$ means that if one variable grows the other decreases. In other words, the expression \[xy\] is constant, to solve the inverse variation questions, as the inverse variation equation is \[y = \dfrac{k}{x}\], in which we need to find the relationship if $y$ varies indirectly as the cube of $x$, hence by understanding the meaning of indirect variation we can find the relationship.

Formula used:
\[y = \dfrac{k}{x}\]
In which, $k$ is the constant of variation and $x$ and $y$ are the variables.

Complete step by step answer:
Let us write the given data: $y$ varies indirectly as the cube of $x$.
We know that the inverse variation sums are solved using the equation:
\[y\alpha k \times \dfrac{1}{x}\]
\[ \Rightarrow \]\[y = \dfrac{k}{x}\]
Indirectly means that if one variable grows the other decreases, so:
\[y = \dfrac{k}{{{x^3}}}\]

Additional information:
When two variables change in inverse proportion it is called as indirect variation. This means that the variables change in the same ratio but inversely. Direct variation means when one quantity changes, the other quantity also changes in direct proportion.

Note:Inverse variation equations are solved using the equation \[y = \dfrac{k}{x}\], two objects or variables vary indirectly means that the product of the two items is a constant. As mentioned, y varies indirectly as the cube of x, hence we get y as:\[y = \dfrac{k}{{{x^3}}}\], hence by the Inverse variation equation we can find the relationship of the given data.