Question : A woman walks 100 yards eastward along a straight shoreline and then swims 30 yards southward into the ocean : 2162255

The rectangular coordinates of a point are given. Find polar coordinates for the point.

42) (√(3), 1)

A) (2, - (5π/6))

B) (2, (5π/6))

C) (2, - (π/6))

D) (2, (π/6))

43) (-3, 2.5) Round the polar coordinates to two decimal places, with θ in radians.

A) (3.91, 2.45)

B) (-3.91, 0.88)

C) (3.91, 0.88)

D) (3.91, -0.88)

44) (0.6, -1.1) Round the polar coordinates to two decimal places, with θ in degrees.

A) (1.25, 57.93°)

B) (1.25, 61.39°)

C) (1.25, -61.39°)

D) (1.25, -57.93°)

Solve the problem.

45) A woman walks 100 yards eastward along a straight shoreline and then swims 30 yards southward into the ocean on a line that is perpendicular to the shoreline. Using her starting point as the pole and the east direction as the polar axis, give her current position (i) in rectangular coordinates and (ii) in polar coordinates. Round the coordinates to the nearest hundredth. Express θ in degrees.

A) (i) (100, 30);

(ii) (11.40, -16.70°)

B) (i) (100, -30);

(ii) (104.40, -16.70°)

C) (i) (100, -30);

(ii) (11.40, -88.28°)

D) (i) (-100, 30);

(ii) (104.40, -88.28°)

46) A fire truck is en route to an address that is 6 blocks east and 11 blocks south of the fire station. Using the fire station as the pole and the east direction as the polar axis, express the fire truck's destination (i) in rectangular coordinates and (ii) in polar coordinates. Round the coordinates to the nearest hundredth. Express θ in degrees.

A) (i) (6, 11);

(ii) (4.12, -61.39°)

B) (i) (6, -11);

(ii) (12.53, -61.39°)

C) (i) (6, 11);

(ii) (4.12, -28.61°)

D) (i) (6, -11);

(ii) (12.53, -28.61°)

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ).

47) x² + 4y² = 4

A) cos² θ + 4sin²θ = 4r

B) 4cos² θ + sin²θ = 4r

C) r²(4cos² θ + sin² θ) = 4

D) r²(cos² θ + 4sin² θ) = 4

48) x² + y² - 4x = 0

A) rcos²θ = 4sinθ

B) r = 4cosθ

C) rsin²θ = 4cosθ

D) r = 4sinθ

49) x² = 4y

A) 4sin²θ = rcosθ

B) rsin²θ = 4cosθ

C) 4cos²θ = rsinθ

D) rcos²θ = 4sinθ

50) y² = 16x

A) sin²θ = 16r²cosθ

B) sin²θ = 16rcosθ

C) rsin²θ = 16cosθ

D) r²sin²θ = 16cosθ

51) xy = 1

A) r²sin2θ = 2

B) rsin2θ = 2

C) 2rsinθcosθ = 1

D) 2r²sinθcosθ = 1

52) 2xy = 3

A) 2rcosθsinθ = 3

B) r²sin2θ = 6

C) r²sin2θ = 3

D) r²cosθsinθ = 6

53) 2x + 3y = 6

A) r(2cosθ + 3sinθ) = 6

B) 2cosθ + 3sinθ = 6r

C) r(2sinθ + 3cosθ) = 6

D) 2sinθ + 3cosθ = 6r

54) x = -3

A) rsinθ = -3

B) rsinθ = 3

C) rcosθ = 3

D) rcosθ = -3

55) y = 5

A) r = 5

B)sinθcosθ = 5

C) rcosθ = 5

D) rsinθ = 5

56) y = x

A) r = sinθ

B) r = cosθ

C) sinθ = -cosθ

D) sinθ = cosθ

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y).

57) r = cosθ

A) (x + y)² = x

B) x² + y² = y

C) x² + y² = x

D) (x + y)² = y

58) r = 1 + 2sinθ

A) x² + y² = √(x² + y²) + 2y

B) x² + y² = √(x² + y²) + 2x

C) √(x² + y²) = x² + y² + 2x

D) √(x² + y²) = x² + y² + 2y

59) r = 10sinθ

A) √(x² + y²) = 10y

B) x² + y² = 10y

C) x² + y² = 10x

D) √(x² + y²) = 10x

60) r = 2(sinθ -cosθ)

A) 2x² + 2y² = y - x

B) 2x² + 2y² = x - y

C) x² + y² = 2x - 2y

D) x² + y² = 2y - 2x

61) r = 5

A) x² + y² = 25

B) x + y = 25

C) x + y = 5

D) x² - y² = 25

62) r = (5/1 +cosθ)

A) y² = 10x - 25

B) y² = 25 - 10x

C) x² = 25 - 10y

D) x² = 10y - 25

63) rsinθ = 10

A) y = 10x

B) y = 10

C) x = 10

D) x = 10y

64) r(1 - 2cosθ) = 1

A) x² + y² = 2 + x

B) x² + y² = 1 + 2x

C) x² + y² = (2 + x)²

D) x² + y² = (1 + 2x)²