- 2072 Asoj · 8 marks
Fit y = a + b√x
Short answer
Transform by X=√x to linear form. Normal equations: 1.87=5a+239.410b, 99.8667=239.410a+13500b. Solving: a=0.1312, b=0.005071. Fitted curve: y=0.1312+0.00507√x.
Fit the following data into y = a + b√x
| X | 500 | 1000 | 2000 | 4000 | 6000 |
|---|---|---|---|---|---|
| Y | 0.20 | 0.33 | 0.38 | 0.45 | 0.51 |
Put X = √x so the model becomes the straight line y = a + bX. Then Σy = na + bΣX and ΣXy = aΣX + bΣX².
| x | y | X = √x | X² (= x) | Xy |
|---|---|---|---|---|
| 500 | 0.2 | 22.361 | 500 | 4.4721 |
| 1000 | 0.33 | 31.623 | 1000 | 10.4355 |
| 2000 | 0.38 | 44.721 | 2000 | 16.9941 |
| 4000 | 0.45 | 63.246 | 4000 | 28.4605 |
| 6000 | 0.51 | 77.460 | 6000 | 39.5044 |
| Σ | 239.410 | 13500 | 99.8667 |
1.87 = 5a + 239.410b
99.8667 = 239.410a + 13500b
Solving: b = (5·99.8667 − 239.410·1.87)/(5·13500 − 239.410²) = 0.005071, a = 0.1312.
Answer: y = 0.1312 + 0.00507√x (check: x = 4000 gives 0.452 against 0.45).
✓ Answer checked by Claude Opus 5.5 (high effort)
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