Other forms of inverse proportion

\(y\) may be inversely proportional to a power, a root or another expression in \(x\). Divide \(k\) by exactly the expression stated.

A power or root of \(y\) can also be the proportional quantity. To name the two quantities in an equation, rearrange it into a constant product: from \(y^2=\dfrac{8}{x}\), \(xy^2=8\), so \(y^2\) is inversely proportional to \(x\).

Why: let \(X=x^2\). Then \(y=\dfrac{k}{X}\), which is ordinary inverse proportion between \(y\) and \(X\). Every method for \(y=\dfrac{k}{x}\) works with \(X\) in place of \(x\).

Method

1. Identify the expression after “inversely proportional to” and write \(y=\dfrac{k}{\text{expression}}\).
2. Find \(k\) by multiplying the given \(y\)-value by the value of the expression.
3. Rewrite the equation with \(k\), then substitute and solve.
4. Check the domain: the expression cannot be \(0\), a square root needs a non-negative value, and taking a square root gives \(\pm\).

Example

The loudness, \(L\) units, of a speaker is inversely proportional to the square of the distance, \(d\) m, from the speaker. At \(3\) m, \(L=40\). Find \(L\) at \(2\) m and the distance at which \(L=10\).

\(\begin{aligned} L&=\dfrac{k}{d^2} \\ 40&=\dfrac{k}{3^2} \\ k&=360 \\ \therefore L&=\dfrac{360}{d^2} \end{aligned}\)

When \(d=2\): \(L=\dfrac{360}{4}=90\).

When \(L=10\): \(d^2=\dfrac{360}{10}=36\), so \(d=6\) m, rejecting \(-6\) because a distance is positive.

Second example: \(y^2\) is inversely proportional to \(x\), and \(y=6\) when \(x=2\), where \(y>0\). Then \(6^2=\dfrac{k}{2}\), so \(k=72\) and \(y^2=\dfrac{72}{x}\). When \(x=8\), \(y^2=9\), so \(y=3\).

Check: a common error is \(k=40\times3=120\), which forgets the square. Moving from \(3\) m to \(6\) m doubles \(d\) and divides \(L\) by \(2^2=4\): \(40\div4=10\).