Final answer:
To convert polar coordinates to Cartesian coordinates, use the formulas x = rcos(θ) and y = rsin(θ). To find distances in the Cartesian system, apply the distance formula. To convert Cartesian coordinates to polar, calculate the radius and use the arc tangent to find the angle.
Step-by-step explanation:
The student is asking how to convert polar coordinates to Cartesian coordinates and calculate distances between points in both coordinate systems. To convert the polar coordinates of a point (r, θ) to Cartesian coordinates (x, y), the following formulas are used:
For the point with polar coordinates (r = 47/3, θ = 5.50 m), we first need to confirm that θ is given in radians. If it is, we apply the formulas directly. If θ is in degrees, we need to convert it to radians first.
To find the distance between two points in the Cartesian coordinate system, we use the distance formula:
- Distance = √((x2 - x1)2 + (y2 - y1)2)
After calculating the Cartesian coordinates for P1 and P2, plug these values into the distance formula to get the distance between them.
For the chameleon example, the distance from the corner of the screen is found using the Pythagorean theorem, given it's coordinates (2.000 m, 1.000 m):
The chameleon's location in polar coordinates requires calculating the radius r as the distance from the origin and determining the angle θ using the arc tangent of y/x.
Similarly, to find the rectangular coordinates of the coins with respect to the well, we compute the x and y using the given polar coordinates, considering the angles relative to the east-west line. The polar coordinates are unchanged as they're already given.