Chapter 9.5, Problem 79E

a.

To determine

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

Answer to Problem 79E

  $r=30$

Explanation of Solution

Given information:

The passengers enter a car at $\left(30,\frac{-\pi }{2}\right)$

Since the passenger enters the car at $r=30$ , which is the directed distance from the pole (center),

Therefore, the polar equation will be:

  $r=30$

which represents a circle with radius = 30 ft

b.

To determine

To find: The coordinate after 15 seconds of rotation.

Answer to Problem 79E

  $\left(30,\frac{5\pi }{6}\right)$

Explanation of Solution

Given information:

The time of rotation is 15 seconds.

Starting from the bottom $\left(30,\frac{-\pi }{2}\right)$ i.e. $\left(30,\frac{3\pi }{2}\right)$

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

  $\Rightarrow \theta =\frac{15s}{45s}\cdot \left(2\pi \right)$

  $\Rightarrow \theta =\frac{2\pi }{3}$

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

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

  $\Rightarrow {\theta }^1=\frac{5\pi }{6}$

Therefore, the coordinate will be:

  $\left(30,\frac{5\pi }{6}\right)$

c.

To determine

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

Answer to Problem 79E

  $\left(-25.98,15\right)$

Explanation of Solution

Given information:

The polar coordinate is $\left(30,\frac{5\pi }{6}\right)$

We know that,

  $x=r\cos \left(\theta \right)$

  $\Rightarrow x=30\cos \left(\frac{5\pi }{6}\right)$

  $\Rightarrow x=30\times \left(-0.866\right)$

  $\Rightarrow x=-25.98$

  $y=r\sin \left(\theta \right)$

  $\Rightarrow y=30\sin \left(\frac{5\pi }{6}\right)$

  $\Rightarrow y=30\times 0.5$

  $\Rightarrow y=15$

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

i.e. $\left(-25.98,15\right)$

This means that the car is 25.98 feet to the left of the center and 15 feet above the center since the center corresponds to the origin.