Chapter 10.2, Problem 49E

$\text{(a)}$

To determine

To find: The slope of the path of the particle at time $t=\pi /2$ .

Answer to Problem 49E

The answer: $-1$

Explanation of Solution

Given information:

Any time $t$ , $0\le t\le 2\pi$ is given parametrically given by,

  $\begin{array}{c}x=e^t\sin t\\ y=e^t\cos t\end{array}$

Calculation:

  $\begin{array}{c}\frac{dx}{dt}=e^t\times \sin (t)+e^t\times \cos (t)\\ \Rightarrow \frac{dy}{dt}=e^t\times \cos (t)-e^t\times \sin (t)\\ \Rightarrow \frac{dy}{dx}=\frac{e^t\times \cos (t)-e^t\times \sin (t)}{e^t\times \sin (t)+e^t\times \cos (t)}\\ \frac{e^{\frac{\pi }{2}}\times \cos \left(\frac{\pi }{2}\right)-e^{\frac{\pi }{2}}\times \sin \left(\frac{\pi }{2}\right)}{e^{\frac{\pi }{2}}\times \sin \left(\frac{\pi }{2}\right)+e^{\frac{\pi }{2}}\times \cos \left(\frac{\pi }{2}\right)}\\ =\frac{-e^{\frac{\pi }{2}}}{e^{\frac{\pi }{2}}}\\ =-1\end{array}$

Therefore, the required slope of the path of the particle at time $t=\pi /2$ is $-1$ .

$\text{(b)}$

To determine

To find: The speed of the particle when $t=1$ .

Answer to Problem 49E

The answer:

Explanation of Solution

Given information:

Any time $t$ , $0\le t\le 2\pi$ is given parametrically given by,

  $\begin{array}{c}x=e^t\sin t\\ y=e^t\cos t\end{array}$

Calculation:

Know that,

  

Then,

  

From part $(a)$ :

  $\begin{array}{c}dx/dt=e^t\cdot (\sin t+\cos t)\\ dy/dt=e^t\cdot (\cos t-\sin t),\text{t}\end{array}$

Then,

  $\begin{array}{c}\frac{dx}{dt}(1)=e^1\cdot (\sin 1+\cos 1)\\ =e\cdot (\sin 1+\cos 1)\\ \frac{dy}{dt}(2)=e^1\cdot (\cos 1-\sin 1)\\ =e\cdot (\cos 1-\sin 1)\end{array}$

According to the previous result,

  

The magnitude of a vector

   is equal to

Then,

  

Therefore, the required speed of the particle when $t=1$ is .

$(c)$

To determine

To find: The distance travelled by the particle.

Answer to Problem 49E

The answer: $\approx 2.430$

Explanation of Solution

Given information:

Any time $t$ , $0\le t\le 2\pi$ is given parametrically given by,

  $\begin{array}{c}x=e^t\sin t\\ y=e^t\cos t\end{array}$

Calculation:

Know that,

  

From part $(b)$ :

  

The distance that the travelled from $t=0$ to $t=1$

Is equal to:

  

Therefore, the required distance travelled by the particle from $t=0$ to $t=1$ is equal to:

  $\approx 2.430$