Question

Two points in space have polar coordinates r1 = 12.5 cm, θ1 = 230º and r2 = 3.3 cm, θ2 = 18º. (a) What are the corresponding Cartesian coordinates of these points? (b) What is the distance between these points?

Answer 1

To convert polar coordinates to Cartesian coordinates, use the formulas: x = r * cos(theta) and y = r * sin(theta). For the given points, the Cartesian coordinates are (x1, y1) = (-7.81 cm, -9.90 cm) and (x2, y2) = (3.17 cm, 0.96 cm). The distance between the points is approximately 15.43 cm.

Step-by-step explanation:

To convert polar coordinates to Cartesian coordinates, we use the formulas:

x = r * cos(theta)
y = r * sin(theta)

(a) For the first point with polar coordinates r1 = 12.5 cm, theta1 = 230º:
x1 = 12.5 * cos(230º) = -7.81 cm
y1 = 12.5 * sin(230º) = -9.90 cm

(b) For the second point with polar coordinates r2 = 3.3 cm, theta2 = 18º:
x2 = 3.3 * cos(18º) = 3.17 cm
y2 = 3.3 * sin(18º) = 0.96 cm

To find the distance between the two points, we use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
distance = sqrt((3.17 - (-7.81))^2 + (0.96 - (-9.90))^2)
distance = sqrt(109.77 + 128.12)
distance = sqrt(237.89)
distance ≈ 15.43 cm

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