- Asked 2/13
- 2075 Baisakh · 8 marks
Fit curve y = a x^b (A)
Short answer
Take log10: log y = log a + b log x. Linear least squares gives log a ≈ -0.2216, b≈3.1734. Thus a=10^{-0.2216}≈0.6003, so the fitted curve is y=0.6003 x^3.1734.
Fit the curve y = ax^b to the following data:
| x | 4 | 5 | 7 | 10 | 11 | 13 |
|---|---|---|---|---|---|---|
| y | 48 | 100 | 294 | 900 | 1210 | 2028 |
Take logarithms (base 10): y = a·xᵇ → log y = log a + b·log x. Put Y = log y, X = log x, A = log a, so Y = A + bX, a straight line fitted by least squares.
| x | y | X = log x | Y = log y | X² | XY |
|---|---|---|---|---|---|
| 4 | 48 | 0.6021 | 1.6812 | 0.3625 | 1.0122 |
| 5 | 100 | 0.6990 | 2.0000 | 0.4886 | 1.3979 |
| 7 | 294 | 0.8451 | 2.4683 | 0.7142 | 2.0860 |
| 10 | 900 | 1.0000 | 2.9542 | 1.0000 | 2.9542 |
| 11 | 1210 | 1.0414 | 3.0828 | 1.0845 | 3.2104 |
| 13 | 2028 | 1.1139 | 3.3071 | 1.2409 | 3.6839 |
| Σ | 5.3015 | 15.4937 | 4.8906 | 14.3447 |
Normal equations (b = B)
ΣY = n·A + B·ΣX → 15.4937 = 6A + 5.3015B
ΣXY = A·ΣX + B·ΣX² → 14.3447 = 5.3015A + 4.8906B
Solving: B = (nΣXY − ΣXΣY)/(nΣX² − (ΣX)²) = 3.1734, A = (ΣY − BΣX)/n = -0.2216.
Then a = 10ᴬ = 10^(-0.2216) = 0.6003, b = 3.1734.
Answer: y = 0.6003 x^3.1734 (check: at x = 10, y ≈ 894.8 against 900).
✓ Answer checked by Claude Opus 5.5 (high effort)
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