Chapter 10.1, Problem 2E

To determine

To find: The equation that relates $x$ and $y$ directly.

Answer to Problem 2E

The equation is $y=\frac{x^2+7}{4}$

Explanation of Solution

Given information:

The equations $x=\sqrt{t-2}$ and

  $y=\frac{t+5}{4}$ for $t$ in the interval $[3,11]$

Calculation:

The given equations

  $\begin{array}{c}x=\sqrt{t-2},\\ y=(t+5)/4,\end{array}$

And

  $t\in [3,11]$

To use the previous result, the parametric curve in the following graph.

  

Thus may infer from the graph that the parametric curve generates a line segment, which indicates that it defines $y$ as a function of $x$ .

From the equation $x=\sqrt{t-2}$ ,

So,

  $\begin{array}{c}{x}^2=t-2\\ \Rightarrow t=x^2+2\end{array}$

Put,

  $t=x^2+2$ in the equation

  $y=(t+5)/4$

So,

  $\begin{array}{c}y=\frac{t+5}{4}\\ =\frac{x^2+2+5}{4}\\ =\frac{x^2+7}{4}\end{array}$

Thus can conclude from the previous findings that the equation connecting $x$ and $y$ is equivalent to:

  $y=\frac{x^2+7}{4}$

Therefore, the required parametric curve defines $y$ as a function of $x$ , while the equation that relates $x$ and $y$ is equal to $y=\frac{x^2+7}{4}$ .