Chapter 11.3, Problem 7QR

To determine

To find: the slop of the curve when $x=3\cos t$ and $y=5\sin t$ for $0\le t\le 2\pi$ .

Answer to Problem 7QR

The slop of the curve is 0.763.

Explanation of Solution

Given information:

  $\begin{array}{l}x=3\cos t\\ y=5\sin t\\ t=2\end{array}$

Formula used:

  $du/dv=\frac{du/dx}{dv/dx}$

Calculation:

Given that theexpression of $x$ is $3\cos t$ for $0\le t\le 2\pi$ .

Take differentiation of $x$ with respect to $t$ .

  $\begin{array}{c}dx/dt=\frac{d}{dt}3\cos t\\ =-3\sin t\end{array}$

Similarly, the given expression of $y$ is $5\sin t$ for $0\le t\le 2\pi$ .

Take differentiation of $y$ with respect to $t$ .

  $\begin{array}{c}dy/dt=\frac{d}{dt}5\sin t\\ =5\cos t\end{array}$

Now, calculate $dy/dx$ . Use the formula $du/dv=\frac{du/dx}{dv/dx}$ ,.

  $\begin{array}{c}dy/dx=\frac{dy/dt}{dx/dt}\\ =\frac{5\cos t}{-3\sin t}\\ =-\frac{5}{3}\cot t\end{array}$

Substitute the value 2 for $t$ .

  $\begin{array}{c}dy/dx=-\frac{5}{3}\cot 2\\ \approx 0.763\end{array}$

Conclusion:

Hence, the required slop of the curve is 0.763.