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8.07

15. find all polar coordinates of point p where p = ordered pair 1 comma pi divided by 3 .

a) (1, pi divided by 3 + 2nπ) or (-1, pi divided by 3 + (2n + 1)π)
b) (1, pi divided by 3 + (2n + 1)π) or (-1, pi divided by 3 + 2nπ)
c) (1, pi divided by 3 + nπ) or (-1, pi divided by 3 + nπ)
d) (1, pi divided by 3 + 2nπ) or (-1, pi divided by 3 + 2nπ)

16. determine two pairs of polar coordinates for the point (3, -3) with 0°≤ θ < 360°.

a) (3 square root of 2 , 315°), (-3 square root of 2 , 135°)
b) (3 square root of 2 , 225°), (-3 square root of 2 , 45°)
c) (3 square root of 2 , 45°), (-3 square root of 2 , 225°)
d) (3 square root of 2 , 135°), (-3 square root of 2 , 315°)

17. the graph of a limacon curve is given. without using your graphing calculator, determine which equation is correct for the graph.
a circular graph with an indentation on the left toward the origin, but not touching the origin

[-5, 5] by [-5, 5]

a) r = 2 + 3 cos θ
b) r = 3 + 2 cos θ
c) r = 2 + 2 cos θ
d) r = 4 + cos θ

18. determine if the graph is symmetric about the x-axis, the y-axis, or the origin.
r = 4 - 4 cos θ

a) no symmetry
b) x-axis only
c) y-axis only
d) origin only

19. a railroad tunnel is shaped like a semiellipse, as shown below.
a semiellipse is shown on the coordinate plane with vertices on the x axis and one point of intersection with the positive y axis.

the height of the tunnel at the center is 58 ft, and the vertical clearance must be 29 ft at a point 21 ft from the center. find an equation for the ellipse.

20. determine if the graph is symmetric about the x-axis, the y-axis, or the origin.
r = 4 cos 5θ

15. Ans: (A)

The general forms of finding all the polar coordinates are:

1) When r >= 0(meaning positive):(r, θ + 2nπ) where, n = integer 
2) When r < 0(meaning negative): (-r, θ + (2n+1)π) where, n = integer 

Since r = +1, -1(ordered pair)
θ(given) = 

When r = +1(r>0):(1, + 2nπ)
When r = -1(r<0):(-1,   + (2n+1)π)

Therefore, the correct option is (A) (1, pi divided by 3 + 2nπ) or (-1, pi divided by 3 + (2n + 1)π)

16. Ans: (A)

In polar coordinates,

Since x = 3, y=-3; therefore,


To find the angle,
tanθ = y/x = -3/3 = -1

=> θ = -45°
=> θ = -45°+360° = 315° (when )

If r =  -r = , then,
θ = -45° + 180° = 135°

Therefore, the correct option is (A) (3 square root of 2 , 315°), (-3 square root of 2 , 135°)

17. Ans: (A)

(Question-17 missing Image is attached below) The general form of the limacon curve is:
r = b + a cosθ

If b < a, the curve would have inner loop.

As you can see in the image attached(labeled Question-17), the limacon curve graph has the inner loop. Therefore, the correct option is (A) r = 2 + 3 cosθ, since b = 2, and a = 3; and the condition b < a (2 < 3) is met.

18.  Ans: (B)

Let's find out!

1. If we replace θ with -θ, we would get:
r = 4 - 4*cos(-θ )
Since, cos(-θ) = +cosθ, therefore,
r = 4 - 4*cos(θ)
Same as the original, therefore, graph is symmetric to x-axis.

2. If we replace r with -r, we would get:
-r = 4 - 4*cos(θ )
r = -4 + 4*cos(θ)
NOT same as original, therefore, graph is NOT symmetric to its origin.

3. If we replace θ with -θ and r with -r, we would get:
-r = 4 - 4*cos(-θ )
Since, cos(-θ) = +cosθ, therefore,
r = -4 + 4*cos(θ)
NOT same as original, therefore, graph is NOT symmetric to y-axis.

Ans: The graph is symmetric to: x-axis only!

19. Ans: 

Explanation:
As the question suggests that it is a horizontal ellipse, therefore, the equation for the horizontal ellipse is:
 -- (A)

Since,
x = 21ft,
y = 29ft,
b = 58ft,
= ? 

Plug-in the values in equation (A),
(A)=> 

=>  = 588

Therefore, the equation becomes,Ans: 

20. Ans: x-axis only

Let's find out!

1. If we replace θ with -θ, we would get:
r = 4*cos(-5θ )
Since, cos(-θ) = +cosθ, therefore,
r = +4*cos(5θ) = Same as original

Therefore, graph is symmetric to x-axis.

2. If we replace r with -r, we would get:
-r = 4*cos(5θ )
r = -4*cos(5θ) = Not same
NOT same as original, therefore, graph is NOT symmetric to its origin.

3. If we replace θ with -θ and r with -r, we would get:
-r = 4*cos(-5θ )
Since, cos(-θ) = +cosθ, therefore,
r = -4*cos(5θ) = Not Same
NOT same as original, therefore, graph is NOT symmetric to y-axis.

Ans: The graph is symmetric to: x-axis only!