Two points are given in polar coordinates by (r,)=(2.00 m, 50.0°) and (r, ) = (5.00 m, 250.0°),respectively.What is the distance between them?
Question
Two points are given in polar coordinates by (r,)=(2.00 m, 50.0°) and (r, ) = (5.00 m, 250.0°),respectively.What is the distance between them?
Solution
To find the distance between two points given in polar coordinates, we first need to convert them into Cartesian coordinates.
The conversion from polar to Cartesian coordinates is given by: x = rcos(θ) y = rsin(θ)
For the first point (r1, θ1) = (2.00 m, 50.0°): x1 = r1cos(θ1) = 2.00 m * cos(50.0°) = 1.28 m y1 = r1sin(θ1) = 2.00 m * sin(50.0°) = 1.53 m
For the second point (r2, θ2) = (5.00 m, 250.0°): x2 = r2cos(θ2) = 5.00 m * cos(250.0°) = -3.83 m y2 = r2sin(θ2) = 5.00 m * sin(250.0°) = -2.71 m
Now, we can find the distance between these two points using the distance formula in Cartesian coordinates, which is given by: d = sqrt((x2 - x1)² + (y2 - y1)²)
Substituting the values we have: d = sqrt((-3.83 m - 1.28 m)² + (-2.71 m - 1.53 m)²) = sqrt((-5.11 m)² + (-4.24 m)²) = sqrt(26.11 m² + 17.98 m²) = sqrt(44.09 m²) = 6.64 m
So, the distance between the two points is approximately 6.64 m.
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