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0606 P13 - Jun 2023 - Q7 - 10 marks
7679
The table shows values of the variables \(x\) and \(y\) which are related by an equation of the form \(y=Ax^b\), where \(A\) and \(b\) are constants.
\(x\)
1.5
2
2.5
3
4
\(y\)
13.8
27.5
46.9
72.6
145
(a) Use the data to draw a straight line graph of \(\ln y\) against \(\ln x\).
(b) Use your graph to estimate the values of \(A\) and \(b\).
(c) Estimate the value of \(x\) when \(y=100\).
Solution
Answer: A straight line graph gives approximately \(A=5\), \(b=2.4\), and \(x\approx3.4\) when \(y=100\).
Transform the given relationship into a straight-line form. The gradient and intercept of the straight line then give the constants in the original model.
Since
\(y=Ax^b,\)
take natural logarithms:
\(\ln y=\ln A+b\ln x.\)
This is in the straight-line form
\(Y=c+mX,\)
where \(Y=\ln y\), \(X=\ln x\), gradient \(m=b\), and intercept \(c=\ln A\).
