Express \(6\sin t+10\cos t\) in the form \(R\sin(t+\alpha)\), where \(R>0\) and \(0<\alpha<\dfrac\pi2\) radians.[3]
The oscillation of a mass attached to a spring can be modelled by \(x=6\sin t+10\cos t\), where \(x\) mm is the distance of the mass from its rest position \(A\) at time \(t\) seconds.
Explain whether the mass can reach a distance of 18 mm from \(A\).[1]
Find the time when the mass first reaches the maximum distance from \(A\).[2]
After how many seconds does the mass first reach a height of 8 mm from \(A\)?[3]
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Answer:(a) \(\sqrt{136}\sin(t+\alpha)\), \(\alpha=\tan^{-1}(5/3)\approx1.03\). (b)(i) No: maximum \(\sqrt{136}\) mm. (ii) \(0.540\) s. (iii) \(1.36\) s.