Question

find two different parametric equations for the given rectangular equation.

$x = y^{\frac{3}{2}}$

let $x = t$. one parametric equation is $x = t, y = t^{\frac{2}{3}}$. (simplify your answer. type your answer us exponential notation.)

let $x = t^{3}$. a second parametric equation is $x = t^{3}, y = \square$. (simplify your answer. type your ans exponential notation.)

Explanation:

Step1: Substitute \(x = t^3\) into the rectangular equation

Given \(x=y^{\frac{3}{2}}\), substitute \(x = t^3\). So we have \(t^3=y^{\frac{3}{2}}\).

Step2: Solve for \(y\)

Using the property of exponents, if \(a^m=b^n\), then \(a = b^{\frac{n}{m}}\) (for \(a,b>0\)). Here, \(m=\frac{3}{2}\) and \(n = 3\). So \(y=(t^3)^{\frac{2}{3}}\).
By the power - of - a - power rule \((a^m)^n=a^{mn}\), we get \(y=t^{3\times\frac{2}{3}}\).

Answer:

\(y = t^{2}\)