Find the distance between two points A(5,60.,0) and B(10,90.,0) where the points are given in Cylindrical coordinates.a)4.19 unitsb)5.19 unitsc)6.19 unitsd)7.19 unitsCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Converting Cylindrical Coordinates to Cartesian Coordinates:
- Given point A: (5, 60°, 0) and point B: (10, 90°, 0)
- Converting cylindrical coordinates to cartesian coordinates:
- For point A: x = 5 * cos(60°), y = 5 * sin(60°), z = 0
- For point B: x = 10 * cos(90°), y = 10 * sin(90°), z = 0

Calculating Distance in Cartesian Coordinates:
- Distance between points A and B in cartesian coordinates:
- d = √[(x_B - x_A)^2 + (y_B - y_A)^2 + (z_B - z_A)^2]
- d = √[(10 * cos(90°) - 5 * cos(60°))^2 + (10 * sin(90°) - 5 * sin(60°))^2 + (0 - 0)^2]
- d = √[(0 - 2.5)^2 + (10 - 8.66)^2 + 0]
- d = √[6.25 + 1.34]
- d = √7.59
- d ≈ 2.76

Converting Distance back to Cylindrical Coordinates:
- Converting the distance from cartesian to cylindrical coordinates:
- r = √(x^2 + y^2) = √(2.5^2 + 8.66^2) ≈ √(6.25 + 74.85) ≈ √81.1 ≈ 9
- θ = arctan(y/x) = arctan(8.66/2.5) ≈ 73.74°
- z = 0
Therefore, the distance between points A(5, 60°, 0) and B(10, 90°, 0) in cylindrical coordinates is approximately 9 units at an angle of 73.74°. Hence, the correct answer is option 'c) 6.19 units'.

Answer

First convert each point which is in cylindrical coordinates to Cartesian coordinates. Now using the formula, distance =

and substituting the values of x, y, and z in it, we get the required answer as 6.19 units. This sum can also be solved using a direct formula to find distance using two points in Cylindrical coordinates.

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