Question

The graphs below show measurements from cubes with different side lengths.

Answer 1

I don't see the graphs — could you upload them or paste the measured data points? Meanwhile, here are the typical things to look for and quick methods to identify which measurement each graph is showing and to get the underlying formula.

Common cube-measure relationships (side length = s)
- Volume: V(s) = s^3 (cubic growth)
- Surface area: A(s) = 6s^2 (quadratic growth)
- Total edge length: L(s) = 12s (linear growth)
- Space diagonal length: d(s) = s·√3 (linear growth)
- Mass (if uniform density): m(s) ∝ s^3 (same shape as volume)

How to identify the type of growth from a graph or two data points
1. Pick two points (s1, y1) and (s2, y2) from the graph.
2. Compute ratio r = y2 / y1 and ratio of side-lengths R = s2 / s1.
3. If r ≈ R^1 the relationship is linear; if r ≈ R^2 it’s quadratic; if r ≈ R^3 it’s cubic.
- More generally, exponent n = log(r)/log(R).
Example: if s doubles (R = 2) and y increases by ≈8 (r ≈ 8), then n = log(8)/log(2) = 3 → cubic.

If you want an explicit fitted formula
- For power-law relationships y = k s^n, compute n as above and then k = y1 / (s1^n).
- For polynomial relationships you can fit with 2–3 points (linear, quadratic, cubic respectively).

Send the graphs or the data points you see and tell me what you need (identify which measurement, find formula, compare growth rates, predict values, etc.), and I’ll analyze them precisely.