Two variables \(x\) and \(y\) are related by the equation \(xy^2=ax+by\). Explain how a straight line graph can be drawn to represent the given equation.[2]
The table shows experimental values of two variables \(x\) and \(y\). It is known that \(x\) and \(y\) are related by the equation \(y=p\mathrm{e}^{-qx}\) where \(p\) and \(q\) are constants.
\(x\)
\(1\)
\(3\)
\(5\)
\(7\)
\(9\)
\(y\)
\(98.2\)
\(65.9\)
\(44.1\)
\(29.6\)
\(19.8\)
On the grid below plot \(\ln y\) against \(x\) and draw a straight line graph.[2]
Use your graph to estimate
the value of \(p\) and of \(q\),[3]
the value of \(y\) when \(x=2\).[1]
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Answer:(a) Plot \(y^2\) against \(y/x\); gradient \(b\), intercept \(a\). (b)(i) See plotted graph. (b)(ii)(a) \(p\approx120,\ q\approx0.200\). (b)(ii)(b) \(y\approx80.6\), depending on the fitted line.