Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 9 - Section 9.5 - Parametric Equations - Exercise Set - Page 1021: 90

Answer

The rectangular equation for the given parametric equations is ${{x}^{\frac{2}{3}}}+{{y}^{\frac{2}{3}}}=1$.

Work Step by Step

Let us consider the provided parametric equations $\begin{align} & x={{\cos }^{3}}t \\ & y={{\sin }^{3}}t \\ \end{align}$ And take cubic roots on both sides of the parametric equations $\begin{align} & {{x}^{\frac{1}{3}}}=\cos t \\ & {{y}^{\frac{1}{3}}}=\sin t \\ \end{align}$ Then, square both sides of the equations $\begin{align} & {{x}^{\frac{2}{3}}}={{\cos }^{2}}t \\ & {{y}^{\frac{2}{3}}}={{\sin }^{2}}t \\ \end{align}$ Then, add the equations $\begin{align} & {{x}^{\frac{2}{3}}}+{{y}^{\frac{2}{3}}}={{\cos }^{2}}t+{{\sin }^{2}}t \\ & =1 \end{align}$ Thus, the rectangular equation for the given parametric equations is ${{x}^{\frac{2}{3}}}+{{y}^{\frac{2}{3}}}=1$.
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