The sensitivity S S\,S of a precision transducer depends on the frequency ω \omega\,ω of pressure oscillations according to the model
S(ω)=(1+kω)−12,∣kω∣<1 S(\omega) = (1 + k\omega)^{-\frac{1}{2}}, \quad |k\omega| < 1 S(ω)=(1+kω)−21,∣kω∣<1where k k\,k is a constant. Given that the binomial expansion of S(ω)S(\omega)S(ω) in ascending powers of ω \omega\,ω up to the term in ω3 \omega^3\,ω3 is
1+0.6ω+Pω2+Qω3 1 + 0.6\omega + P\omega^2 + Q\omega^3 1+0.6ω+Pω2+Qω3(i) find the value of kkk,
(ii) find the value of the constant P P\,P and the constant QQQ.
Use the expansion to find an approximate value to 10.88\displaystyle \frac{1}{\sqrt{0.88}}0.881.
Show your working and give your answer to 6 decimal places.
