Question

PLEASE HELP ASAP, PRIORITIZED HELP WOULD BE APPRECIATED
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

Answer 1

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) = $\frac{ \pi }{3}$

When r = +1(r>0):(1, $\frac{ \pi }{3}$ + 2nπ)
When r = -1(r<0):(-1, $\frac{ \pi }{3}$  + (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,$r = \sqrt{x^2 + y^2}$

Since x = 3, y=-3; therefore,
$r = \sqrt{(3)^2 + (-3)^2} = 3 \sqrt{2}$

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

=> θ = -45°
=> θ = -45°+360° = 315° (when $r = 3 \sqrt{2}$)

If r =  -r = $3 \sqrt{2}$, 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: $\frac{x^2}{588} + \frac{y^2}{3364} = 1$

Explanation:
As the question suggests that it is a horizontal ellipse, therefore, the equation for the horizontal ellipse is:
$\frac{x^2}{a^2} + \frac{y^2}{b^2} =1$ -- (A)

Since,
x = 21ft,
y = 29ft,
b = 58ft,
$a^2$ = ? 

Plug-in the values in equation (A),
(A)=> $\frac{441}{a^2} + \frac{841}{3364} = 1$

=> $a^2$ = 588

Therefore, the equation becomes,Ans: $\frac{x^2}{588} + \frac{y^2}{3364} = 1$

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!

Answer 2

Hello,
Please, see the attached files.
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