---
title: "Fit y = a + b√x"
subject: "Numerical Method"
chapter: 4
key: "semester_4/numerical_method:fit-y-a-b-sqrt-x"
source: https://www.bctengineer.com/q/semester_4/numerical_method/fit-y-a-b-sqrt-x
site: BCT Engineering
---

# Fit y = a + b√x

Numerical Method · Chapter 4: Interpolation

- Asked in: 2072 Asoj, 8 marks
- Video: [Curve fitting method by the method of Least square | Curve Fitting parabola](https://www.youtube.com/watch?v=-UJr1XjyfME) — Civil learning online
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/fit-y-a-b-sqrt-x
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-4

## Question

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 |

## 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.

## Answer

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 |

```text
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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is y = a + b√x made linear?
   A. Put Y = 1/y
   B. Take log of y
   C. Put X = x^2
   D. Put X = √x so y = a + bX
   Answer: D. With X = √x the model is a straight line in X. (Key point: Substitution)

2. Which are the normal equations?
   A. Σy = bΣX only
   B. Σy = na + bΣX and ΣXy = aΣX + bΣX^2
   C. Σy = a + b and ΣXy = ab
   D. Σy = na + bΣx and Σxy = aΣx
   Answer: B. Standard linear least squares in X. (Key point: Normal equations)

3. Why does the X² column equal x?
   A. Because b = 1
   B. Because X = √x so X^2 = x
   C. Because x is squared
   D. It is a mistake
   Answer: B. (√x)^2 = x, so ΣX^2 = Σx = 13500. (Key point: Use of X^2)

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