Two points in a plane have polar coordinates (2.40 m, 40.0°) and (3.50 m, 110.0°).
(a) Determine the Cartesian coordinates of these points.
(2.40 m, 40.0°)
x = m
y = m
(3.50 m, 110.0°)
x = m
y = m
(b) Determine the distance between them.
m
a. P1: 2.4*Cos40 + 2.4*sin40 = 1.84 + 1.54i.
P1 = (x1,y1) = (1.84,1.54).
P2 = (x2,y2) = (-1.20,3.29).
b. d^2 = (x2-x1)^2 + (y2-y1)^2.
d^2 = (-1.2-1.84)^2 + (3.29-1.54)^2 = 12.3
d = 3.51 m.
Henry ##$$-#-#°£¥¢¥
You forget C
To convert polar coordinates to Cartesian coordinates, you can use the following formulas:
x = r * cosθ
y = r * sinθ
where r is the magnitude (distance) from the origin to the point, and θ is the angle between the positive x-axis and the line connecting the origin to the point.
Let's apply these formulas to the given points:
For the point (2.40 m, 40.0°):
x = 2.40 m * cos(40.0°) ≈ 1.8321 m
y = 2.40 m * sin(40.0°) ≈ 1.5394 m
So, the Cartesian coordinates of this point are approximately (1.8321 m, 1.5394 m).
For the point (3.50 m, 110.0°):
x = 3.50 m * cos(110.0°) ≈ -1.1634 m
y = 3.50 m * sin(110.0°) ≈ 3.2347 m
So, the Cartesian coordinates of this point are approximately (-1.1634 m, 3.2347 m).
To calculate the distance between these two points, you can use the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of the two points into the formula, we get:
d = √((1.8321 m - (-1.1634 m))^2 + (1.5394 m - 3.2347 m)^2)
= √((2.9955 m)^2 + (-1.6953 m)^2)
= √(8.973 m^2 + 2.8771 m^2)
= √11.8501 m^2
≈ 3.4415 m
So, the distance between the two points is approximately 3.4415 m.