Chapter 10, Problem 66RE

To determine

Tofind:the rectangular equation and graph the curve using parametric equation.

Answer to Problem 66RE

The rectangular equation is ${\text{ y=e}}^{\text{3x}}$ and the graph here will be of exponential.

Explanation of Solution

Given:

  ${\text{x=lnt, y=t}}^3$ ; $\text{t > 0}$ .

Concept used:

The parametric equation has equation for each variable whereas rectangular equation has one equation

In rectangular equation it composed of variable like $\text{x and y}$ which can be graphed on a regular cartesian plane.

To convert parametric to rectangular $\left(\text{t}\right)$ should be eliminate so that function will be in the form of $\text{x and y}$ which consist of only one independent variable.

Calculation:

According to the given parametric equation:

  $\text{x=lnt}$

  ${\text{3x=3lnt=lnt}}^{\text{3}}$

  ${\text{3x=lnt}}^{\text{3}}$ …………… $\left(\text{i}\right)$

  ${\text{y=t}}^3$ …………….. $\left(\text{ii}\right)$

So, putting $\left(\text{ii}\right)$ equation $\left(\text{i}\right)$ :

  $\text{3x=ln y}$

  ${\text{ y=e}}^{\text{3x}}$

Which is exponential equation.

The equation is in the form of exponential.

So, the graph will be exponential as shown below:

The graph of the equation ${\text{ y=e}}^{\text{3x}}$ :

  

Here from the graph:

At $\text{x=0; y=1}\text{.}$

Hence, the rectangular equation is ${\text{ y=e}}^{\text{3x}}$ and the graph here will be of exponential.