Question

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=t^{3}, \quad y=3 \ln t $$

Short answer

The corresponding rectangular equation for the given parametric equations is \(y= 3 \ln \sqrt[3]{x}\).

Step-by-step solution

Identify Parameters

First, identify the equations for x and y in term of the parameter t. Here, \(x= t^{3}\) and \(y= 3 \ln t\).

Graphing The Parametric Equations

Plot the points based on each equation. Utilize a graphing utility tool for plotting points. Indicate the orientation of the curve, which shows the direction in which t is increasing.

Convert Parametric Equations To Rectangular Form

You can convert the parametric equations into rectangular form by eliminating the parameter, t. To eliminate t from our equations, we will convert both to an equation in terms of t and set them equal to each other. Given the equation for x, we can write \(t = \sqrt[3]{x}\). Substitute this in place of t in the equation for y, which becomes \(y= 3 \ln \sqrt[3]{x}\).