Final answer:
The Cartesian coordinates are obtained from polar coordinates by using specific formulas based on trigonometric functions. For each set of given polar coordinates, we calculate the corresponding x and y values in meters using cos() and sin() functions, respectively. The distance between two Cartesian points is determined by the distance formula.
Step-by-step explanation:
To find the Cartesian coordinates from polar coordinates, we use the following transformations:
- x = r × cos(θ)
- y = r × sin(θ)
For problem 42, the polar coordinates are given as (47/3, 5.50 m). To find the Cartesian coordinates, we calculate:
- x = (47/3) × cos(5.50)
- y = (47/3) × sin(5.50)
Apply these calculations, and we find our x and y values in meters.
For problem 43, the points have polar coordinates P₁ (2.500 m, π/6) and P₂ (3.800 m, 27/3). To find their Cartesian coordinates:
- P₁: x = 2.500 × cos(π/6), y = 2.500 × sin(π/6)
- P₂: x = 3.800 × cos(27/3), y = 3.800 × sin(27/3)
Then calculate the distance using the distance formula √[(x₂ - x₁)² + (y₂ - y₁)²].
For problem 45, points A and B are given in Cartesian coordinates. We need to find the distance between them using the distance formula and then convert each point to polar coordinates using the transformations mentioned earlier.