Answer with explanation:
We are given a parametric equation as:
\(x=3 \sin^3 t\)
and \(y=3 \cos^3 t\)
Hence, we can represent our equation as:
\(\sin^3 t=\dfrac{x}{3}\\\\\\\sin t=(\dfrac{x}{3})^{\dfrac{1}{3}}\\\\\\Hence,\\\\\sin^2 t=(\dfrac{x}{3})^{\dfrac{2}{3}}\\\\and\ similarly\\\\\cos^3 t=\dfrac{y}{3}\\\\\cos t=(\dfrac{y}{3})^{\dfrac{1}{3}}\\\\Hence,\\\\\cos^2 t=(\dfrac{y}{3})^{\dfrac{2}{3}}\)
As we know that:
\(\cos^2 t+\sin^2 t=1\)
Hence, on putting the value in the formula we get the equation in rectangular coordinates as:
\((\dfrac{x}{3})^{\dfrac{2}{3}}+(\dfrac{y}{3})^{\dfrac{2}{3}}=1\)
Hence, this is a equation of a ASTROID.
