Chapter 1.4, Problem 42E

To determine

To find: The correct option that represents the graph of curve of the parametric curve $x=3t$ and $y=2t$ in the interval $t\ge 1$.

Answer to Problem 42E

The correct choice is (E).

Explanation of Solution

Given information: The parametric curves are $x=3t$ and $y=2t$ in the interval $t\ge 1$.

Calculation:

Solve the equation $x=3t$ for $t$.

  $t=\frac{x}{3}$

Substitute $\frac{x}{3}$ for $t$ in the equation $y=2t$.

  $\begin{array}{c}y=2\left(\frac{x}{3}\right)\\ =\frac{2}{3}x\end{array}$

This is the equation of line and the value of $t$ is restricted to $t\ge 1$.

Substitute 1 for $t$ in the equation $x=3t$ and $y=2t$.

  $\begin{array}{c}x=3\left(1\right)\\ =3\\ y=2\left(1\right)\\ =2\end{array}$

As $t$ increases $x$ and $y$ both increases without bound so it has no terminal point. So, the graph is a ray because it has a starting point but no terminal point.

Therefore, the correct choice is (E).