Practice Scaling Laws in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Relationships describing how a quantity changes when the size or scale of a system is multiplied by a factor, often expressed as power laws.
When you double the length of a cube, its volume grows by 2^3 = 8. Scaling laws reveal how fast quantities grow — they often explain why small and large things behave so differently.
Example 1
easyIf you scale a square's side length by a factor of 3, by what factor does the area change? State the general scaling law.
Example 2
mediumA sphere of radius r has volume V = \frac{4}{3}\pi r^3. If the radius doubles, by what factor does the volume increase? Generalise to a scaling factor k.
Example 3
easyIf the radius of a circle doubles, by what factor does the circumference change? What about the area?
Example 4
mediumA population model gives P(t) = P_0 e^{rt}. If r is doubled (growth rate doubles), by what factor does P(T) change for a fixed time T?