Chapter 1.4, Problem 3E

To determine

To match: The parametric equations with its appropriate graph, write the approximate dimensions of the viewing window and a parameter interval that traces the curve exactly once.

Answer to Problem 3E

The correct match of the graph for given parametric equation is (d), the dimension of the window is $\left[-10,10\right]$ by $\left[-10,10\right]$ and parameter interval that traces the curve exactly once is $0\le t\le 2\pi$.

Explanation of Solution

Given information: The parametric equations are $x=7\sin t-\sin \left(7t\right)$ and $y=7\cos t-\cos \left(7t\right)$.

Calculation:

Use the following step to graph the parametric equations by graphing calculator.

Step 1: Press the $\boxed{\text{MODE}}$ button, under FUNC and choose PAR option.

Step 2: Press $\boxed{\text{Y=}}$ , and set the parametric equations under ${\text{X}}_{1\text{T}}$ and ${\text{Y}}_{1\text{T}}$.

Step 3: Press $\boxed{\text{GRAPH}}$ to obtain the graph of the parametric equations and the window can be set to $\left[-10,10\right]$ by $\left[-10,10\right]$ as shown below.

  

Figure (1)

First find the starting point at $t=0$ to find the period of the function as shown below.

Substitute 0 for $t$ in the parametric equation $x=7\sin t-\sin \left(7t\right)$.

  $\begin{array}{c}x=7\sin \left(0\right)-\sin \left(7\left(0\right)\right)\\ =0\end{array}$

Substitute 0 for $t$ in the parametric equation $y=7\cos t-\cos \left(7t\right)$.

  $\begin{array}{c}y=7\cos \left(0\right)-\cos \left(7\left(0\right)\right)\\ =7-1\\ =6\end{array}$

Find the next value of $t$ for which the coordinates of the curve are $\left(0,6\right)$ . After $t=0$ , the next value of $t$ for which $y=6$ is $t=2\pi$.

Substitute $2\pi$ for $t$ in the parametric equation $y=7\cos t-\cos \left(7t\right)$.

  $\begin{array}{c}y=7\cos \left(2\pi \right)-\cos \left(7\left(2\pi 0\right)\right)\\ =7\left(1\right)-1\\ =7-1\\ =6\end{array}$

Substitute $2\pi$ for $t$ in the parametric equation $x=7\sin t-\sin \left(7t\right)$.

  $\begin{array}{c}x=7\sin \left(2\pi \right)-\sin \left(14\pi \right)\\ =0\end{array}$

The interval is $0\le t\le 2\pi$.

Therefore, the correct match of the graph for given parametric equation is (d), the dimension of the window is $\left[-10,10\right]$ by $\left[-10,10\right]$ and parameter interval that traces the curve exactly once is $0\le t\le 2\pi$.