N2018 P2 Q9

N2018 P2 Q9

Junior College 2
12 marks
Free

Many electronic devices need a fan to keep them cool. In order to maximise the lifetime of such fans, the speed they run at is reduced when conditions allow. Running a fan at a lower speed reduces the power required. The following table gives details, for a particular type of fan, of the power required (\(P\) watts) at different fan speeds (\(R\) revolutions per minute).

Fan speed (\(R\))\(3600\)\(4500\)\(5400\)\(6300\)\(7200\)\(8100\)\(9000\)\(9900\)
Power (\(P\))\(0.22\)\(0.34\)\(0.52\)\(0.78\)\(1.06\)\(1.48\)\(2.04\)\(2.64\)
  1. Draw a scatter diagram of these data. Use your diagram to explain whether the relationship between \(P\) and \(R\) is likely to be well modelled by an equation of the form \(P=aR+b\), where \(a\) and \(b\) are constants.[2]
  2. By calculating the relevant product moment correlation coefficients, determine whether the relationship between \(P\) and \(R\) is modelled better by \(P=aR+b\) or by \(P=aR^2+b\). Explain how you decide which model is better, and state the equation in this case.[5]
  3. Use your equation to estimate the speed of the fan when the power is \(0.9\) watts. Explain whether your estimate is reliable.[2]
  4. Use your equation to estimate the power used when the speed of the fan is \(3300\) revolutions per minute. Explain whether your estimate is reliable.[2]
  5. Re-write your equation from part (ii) so that it can be used when the speed of the fan, \(R\), is given in revolutions per second.[1]

Default solution

(i) Plot the eight \((R,P)\) pairs from the table. The points curve upwards, so a straight-line model in \(R\) is less suitable.

(ii) \(r_{P,R}=0.969291\), whereas \(r_{P,R^2}=0.993039\). The latter is nearer \(1\).

Regressing \(P\) on \(R^2\) gives \(P=2.845744476\times10^{-8}R^2-0.282607611\), or \(P\approx2.85\times10^{-8}R^2-0.283\).

(iii) \(0.9=2.845744476\times10^{-8}R^2-0.282607611\Rightarrow R\approx6446\text{ rpm}\approx6450\text{ rpm}\). This interpolates within the observed speed range and is reasonably reliable.

(iv) \(R=3300:\quad P\approx0.0273\text{ W}\). This extrapolates below the observed range; the estimate is unreliable.

(v) For \(R\) revolutions per second, the original speed is \(60R\) rpm. \(P=2.845744476\times10^{-8}(60R)^2-0.282607611\approx1.02\times10^{-4}R^2-0.283\).

Answer:(ii) \(P\approx2.85\times10^{-8}R^2-0.283\); (iii) \(6450\text{ rpm}\), reliable; (iv) \(0.0273\text{ W}\), unreliable; (v) \(P\approx1.02\times10^{-4}R^2-0.283\)

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