Linear Law - Plotting Linearised Data and Lines of Best Fit
Linear Law - Plotting Linearised Data and Lines of Best Fit
The diagram below shows a stream of water flowing out from the end of a drainage pipe into a larger canal. The vertical height, \(y\) metres, of the stream above the surface level of the canal is measured at various horizontal distances, \(x\) metres, from the end of the pipe. The results are recorded in the table:
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
|---|---|---|---|---|---|---|
| \(y\) | \(1.20\) | \(1.14\) | \(1.04\) | \(0.90\) | \(0.86\) | \(0.50\) |
The motion of the water stream can be represented by the equation \(y = P + Qx^2\), where \(P\) and \(Q\) are constants. However, one of the \(y\)-values in the table have been recorded incorrectly.
- Using graph paper, plot \(y\) against \(x^2\) and draw a straight line graph.
- Based on your graph, identify which value of \(y\) is incorrect and estimate the correct value.
- estimate the values of \(P\) and \(Q\).
- Explain the meaning of the value of \(P\) in the context of the water stream.
Solution:
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Answer:(i) Fitted line \(y\approx1.22-0.020x^2\). (ii) Incorrect \(0.86\) at \(x=5\); corrected height approximately \(0.72\) m. (iii) \(P\approx1.22,\ Q\approx-0.020\). (iv) \(P\) is the stream height at the pipe, approximately \(1.22\) m above the canal.
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