Chapter 9.5, Problem 88E

a.

To determine

To find: The polar equations that models the possible positions of a passenger car.

Answer to Problem 88E

  $r=35$

Explanation of Solution

Given information:

The passengers enter a car at $\left(35,\frac{-\pi }{2}\right)$

Since the passenger enters the car at $r=35$ , which is the directed distance from the pole (center),

Therefore, the polar equation will be:

  $r=35$

which represents a circle with radius = 35 feet

b.

To determine

To find: The coordinate after 15 seconds of rotation.

Answer to Problem 88E

Explanation of Solution

Given information:

The time of rotation is 15 seconds.

Starting from the bottom $\left(35,\frac{-\pi }{2}\right)$ i.e. $\left(35,\frac{3\pi }{2}\right)$

The Ferris wheel will rotate at an angle of $\theta$

  $\Rightarrow \theta =\frac{15s}{60s}\cdot \left(2\pi \right)$

  $\Rightarrow \theta =\frac{\pi }{2}$

So, therefore the position’s angle in standard position is:

  $\Rightarrow {\theta }^1=\frac{3\pi }{2}-\frac{\pi }{2}$

  $\Rightarrow {\theta }^1=\pi$

Therefore, the coordinate will be:

  $\left(35,\pi \right)$

c.

To determine

To find: To convert the point obtained in (b) into rectangular coordinates.

Answer to Problem 88E

  $\left(-35,0\right)$

Explanation of Solution

Given information:

The polar coordinate is $\left(35,\pi \right)$

We know that,

  $x=r\cos \left(\theta \right)$

  $\Rightarrow x=35\cos \left(\pi \right)$

  $\Rightarrow x=35\times \left(-1\right)$

  $\Rightarrow x=-35$

  $y=r\sin \left(\theta \right)$

  $\Rightarrow y=35\sin \left(\pi \right)$

  $\Rightarrow y=35\times 0$

  $\Rightarrow y=0$

Therefore, the rectangular coordinate is $\left(x,y\right)$ :

i.e. $\left(-35,0\right)$

This means that the car is 35 feet to the left of the center and at the level of the center since the center corresponds to the origin.