A woman walks 200 yards west along a straight shoreline and then swims 50 yards north 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 polar coordinates. Round the coordinates to the nearest hundredth. Express θ in degrees

Question

A woman walks 200 yards west along a straight shoreline and then swims 50 yards north 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 polar coordinates. Round the coordinates to the nearest hundredth. Express θ in degrees

Answer 1

r^2 = 200^2 + 50^2 = 42500

r = √42500 = 50√17

tanØ = -50/200 = -1/4
Ø = 165.96°

so using the (r,Ø) notation , her position is
(50√17, 165.96°)

Answer 2

To determine the current position of the woman in polar coordinates, we can use the distance and angle from her starting point. Let's break down the information given:

- The woman walks 200 yards west along the shoreline, which means she moves directly opposite to the east direction.
- Then she swims 50 yards north into the ocean, which is perpendicular to the shoreline.

Since we are using her starting point as the pole and the east direction as the polar axis, we can form a right triangle where the shoreline is the adjacent side and the swim in the ocean is the opposite side. To find the polar coordinates, we need to determine the hypotenuse (distance from the pole) and the angle (θ) formed.

Using the Pythagorean theorem, we can calculate the hypotenuse (distance):

distance^2 = adjacent^2 + opposite^2
distance^2 = 200^2 + 50^2
distance^2 = 40,000 + 2,500
distance^2 = 42,500

Taking the square root of both sides, we find:

distance ≈ √42,500
distance ≈ 206.15 yards

The distance from her starting point to her current position is approximately 206.15 yards.

To determine the angle (θ), we can use trigonometric functions. In this case, we have the opposite side (50 yards) and the adjacent side (200 yards) of the right triangle.

tan(θ) = opposite/adjacent
tan(θ) = 50/200
tan(θ) = 0.25

Taking the arctan (inverse tangent) of both sides, we find:

θ ≈ arctan(0.25)
θ ≈ 14.04 degrees

Therefore, the woman's current position in polar coordinates is approximately (206.15 yards, 14.04 degrees) when using her starting point as the pole and the east direction as the polar axis.