Final answer:
The conversion from polar to Cartesian coordinates involves trigonometry, and the distance between two points on a plane is calculated using the Pythagorean theorem. In the given exercises, Cartesian coordinates are calculated from given polar coordinates, and distances are determined and rounded as necessary.
Step-by-step explanation:
Converting between polar and Cartesian coordinates requires the use of trigonometry. The Cartesian coordinates (x, y) are related to polar coordinates (r, θ) by the formulas x = r × cos(θ) and y = r × sin(θ). Additionally, the distance between two points in the Cartesian plane is found using the distance formula derived from Pythagoras' theorem.
Let's work on the example of points P₁ (2.500 m, π/6) and P₂ (3.800 m, 27π/3). First, convert them to Cartesian coordinates:
- P₁(x₁, y₁): x₁ = 2.500 m × cos(π/6), y₁ = 2.500 m × sin(π/6)
- P₂(x₂, y₂): x₂ = 3.800 m × cos(27π/3), y₂ = 3.800 m × sin(27π/3)
Next, use the distance formula:
√((x₂ - x₁)² + (y₂ - y₁)²)
After calculating, round the result to the nearest centimeter to determine the distance between the points.
For the chameleon on a screen, the distance from the corner is simply the magnitude of the position vector, or √(x² + y²). Its polar coordinates can be found by determining the angle θ = atan2(y, x) and the radial distance r = √(x² + y²).