Three particles travel along parametric paths on : Curve Curve Curve Rank the three…

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Three particles travel along parametric paths on 0≤t≤π0\le t\le \pi: Curve A:  xA(t)=t,  yA(t)=sin⁡tA:\; x_A(t)=t,\; y_A(t)=\sin t Curve B:  xB(t)=t,  yB(t)=2sin⁡tB:\; x_B(t)=t,\; y_B(t)=2\sin t Curve C:  xC(t)=t,  yC(t)=12sin⁡(2t)C:\; x_C(t)=t,\; y_C(t)=\tfrac{1}{2}\sin(2t) Rank the three curves from longest to shortest on the interval [0,π][0,\pi]. Give a complete justification without computing any definite integrals. (a) Derive an explicit expression for the speed of each curve. (b) For every tt in [0,π][0,\pi], compare the speeds of curves AA and BB. Provide algebraic or trigonometric reasoning. (c) Compare the speeds of curves AA and CC for tt in [0,π][0,\pi]. Address any sub-intervals where the ordering might change, and provide supporting work. (d) Combine your findings from parts (b) and (c) to place the three arc lengths in order from greatest to least, and justify your ranking.

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