Linear Law - Predictions from Linearised Models
Linear Law - Predictions from Linearised Models
The table below shows experimental values of two variables \(x\) and \(y\) which are known to be related by an equation of the form \(xy = a\sqrt{x^3} + b\sqrt{x}\), where \(a\) and \(b\) are constants.
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
|---|---|---|---|---|---|
| \(y\) | \(1.1\) | \(3.4\) | \(4.9\) | \(6.1\) | \(7.1\) |
- Plot the graph of \(\frac{y}{\sqrt{x}}\) against \(\frac{1}{x}\) for the given data and use it to estimate the value of \(a\) and of \(b\).
- Use your graph to estimate the value of \(x\) when \(\frac{y}{\sqrt{x}} = 2.7\).
- By adding a suitable straight line to your graph, find the solution of the simultaneous equations \(xy = a\sqrt{x^3} + b\sqrt{x}\) and \(xy = 2\sqrt{x}\).
Solution:
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Answer:(i) \(a\approx3.7,\ b\approx-2.6\) (ii) \(x\approx2.6\) (iii) Additional line \(Y=2X\); positive solution \((x,y)\approx(1.24,1.79)\). Original equations also admit \((0,y)\) for every real \(y\).
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