Final answer:
To find the distance between two Cartesian points, the distance formula is applied. Then, to convert Cartesian coordinates to polar coordinates, the radius and angle are calculated using the radius formula and the atan2 function.
Step-by-step explanation:
To find the distance between the points A(2.00 m, -4.00 m) and B(-3.00 m, 3.00 m), you can use the distance formula:
D = √[(x_2 - x_1)^2 + (y_2 - y_1)^2]
Inserting the coordinates into the formula, we get:
D = √[(-3.00 - 2.00)^2 + (3.00 - (-4.00))^2]
D = √[(-5.00)^2 + (7.00)^2]
D = √[25.00 + 49.00]
D = √[74.00]
D = 8.60 m
Next, to find their polar coordinates, you convert their Cartesian coordinates using the polar coordinate transformations:
r = √[(x^2 + y^2)]
θ = atan2(y, x)
Doing this for point A:
r_A = √[(2.00)^2 + (-4.00)^2] = √[4.00 + 16.00] = √[20.00]
θ_A = atan2(-4.00, 2.00) = atan2(-2, 1)
And for point B:
r_B = √[(-3.00)^2 + (3.00)^2] = √[9.00 + 9.00] = √[18.00]
θ_B = atan2(3.00, -3.00) = atan2(-1, -1)
The exact polar angles would depend on the quadrant in which each point lies and the specific function 'atan2' from a programming language or calculator, which takes into account the signs of both arguments to return the correct angle.