Direct Proportion to a Power

Statement

Sometimes a variable is directly proportional not just to another variable, but to a power of that variable. In such cases we write:

\[ y \propto x^k \quad \Rightarrow \quad y = kx^k \]

Here, \(k\) is a constant of proportionality, and the exponent \(k\) tells us how the variable grows in relation to \(x\).

Why it’s true

• Direct proportion to a power means when \(x\) changes, \(y\) changes by a fixed multiple of \(x^k\).
• If \(k = 1\), this reduces to simple direct proportion.
• If \(k = 2\), then doubling \(x\) makes \(y\) increase by a factor of \(2^2 = 4\).
• Similarly, if \(k = 3\), doubling \(x\) makes \(y\) increase by \(2^3 = 8\).

Recipe (how to use it)

1. Write \( y = kx^k \).
2. Use a known pair of values of \(x\) and \(y\) to calculate \(k\).
3. Use the equation to calculate other values of \(y\) when \(x\) changes.

Spotting it

Look for key phrases such as “\(y\) is proportional to the square of \(x\)” (\(k = 2\)), “\(y\) is proportional to the cube of \(x\)” (\(k = 3\)), or “\(y\) is proportional to the square root of \(x\)” (\(k = \tfrac{1}{2}\)).

Common pairings

• Area of a circle \(A \propto r^2\).
• Volume of a sphere \(V \propto r^3\).
• Time period of a pendulum \(T \propto \sqrt{L}\).
• Energy in physics problems, often proportional to square of speed or current.

Mini examples

1. Given: \(y \propto x^2\), and \(y = 18\) when \(x = 3\). Find: the formula. Answer: \(18 = k \times 3^2 \Rightarrow k = 2\). So \(y = 2x^2\).
2. Given: \(y \propto \sqrt{x}\), and \(y = 10\) when \(x = 25\). Find: \(k\). Answer: \(\sqrt{25} = 5\), so \(10 = k \times 5 \Rightarrow k = 2\). So \(y = 2\sqrt{x}\).

Pitfalls

• Forgetting that the power can be a fraction (square root, cube root).
• Mixing up direct proportion with inverse proportion (\(y \propto 1/x^k\)).
• Failing to compute powers correctly (e.g., squaring vs doubling).

Exam strategy

• Identify the correct power relationship from the wording.
• Always find \(k\) using given values before substitution.
• Check whether the answer should be left exact (e.g., surd form) or as a decimal.

Summary

Direct proportion to a power generalises the idea of proportionality. Instead of \(y\) being proportional to \(x\), it is proportional to \(x^k\). This appears in many real-world situations, from geometry (areas and volumes) to science (laws involving squares and roots). Recognising the correct power is the key step in solving such problems.