Inverse Proportion to a Power

Statement

A quantity is in inverse proportion to a power if it decreases as another variable increases, with the relationship involving a power of the variable. Mathematically:

\[ y \propto \tfrac{1}{x^k} \quad \Rightarrow \quad y = \tfrac{k}{x^n} \]

Here, \(n\) is the power of \(x\), and \(k\) is a constant of proportionality.

Why it’s true

• When one variable decreases as a power of another variable increases, their product involving that power remains constant.
• For example, in physics, gravitational force \(F\) is inversely proportional to the square of distance: \(F \propto \tfrac{1}{r^2}\).
• This reflects the fact that the influence spreads over an area (or volume), which grows as a power of distance.

Recipe (how to use it)

1. Write the formula as \(y = \tfrac{k}{x^n}\).
2. Use known values of \(x\) and \(y\) to find the constant \(k\).
3. Substitute into the equation to find unknown values of \(y\) or \(x\).
4. Remember: \(y \times x^n = k\) is constant.

Spotting it

Problems often say “\(y\) varies inversely with the square (or cube, etc.) of \(x\).” This signals inverse proportion to a power.

Common pairings

• Physics: Inverse-square laws (gravity, light intensity, electrostatics).
• Mathematics: Hyperbolic and rational functions.
• Engineering: Sound intensity and radiation spread.

Mini examples

1. Given: \(y\) varies inversely with \(x^2\). If \(y=20\) when \(x=2\), find \(y\) when \(x=5\). Solution: \(k=yx^2=20×2^2=80.\) For \(x=5\), \(y=80/25=3.2.\)
2. Given: Intensity of light \(I \propto 1/d^2\). If intensity is 100 at distance 2 m, what is it at 10 m? Solution: \(k=100×2^2=400.\) At 10 m, \(I=400/100=4.\)

Pitfalls

• Forgetting the power: Always check whether it’s inverse square, cube, etc.
• Confusing with simple inverse proportion: That’s only \(y=k/x\) (power of 1).
• Arithmetic errors: Be careful with squaring/cubing before dividing.

Exam strategy

• Write \(y \times x^n = k\) at the start.
• Use the given pair of values to calculate \(k\).
• Substitute into the formula to find the unknown.
• Double-check powers and units if it’s a physics context.

Summary

Inverse proportion to a power means one quantity decreases as the power of another increases. Formula: \(y=k/x^n\). Typical examples include inverse-square laws in physics. Remember: always multiply \(y\) by \(x^n\) to get the constant.