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Sagot :
Answer:
R = 73.34 m, θ = 37º North of West
Explanation:
In this exercise you are asked to find the distance traveled.
The best way to do this is with a Cartesian coordinate system, let's use a system where the x axis coincides with the West - East direction), to decompose the trajectories
First trajectory
distance d₁ = 27 m
Direction 32 East of North, this angle is measured from the vertical direction (y-axis)
cos 32 = y₁ / d₁
sin 32 = x₁ / d₁
y₁ = d₁ cos 32
x₁ = d₁ sin 32
y₁ = 27 cos 32 = 22.90 m
x₁ = 27 sin 32 = 14.31 m
second trajectories
distance d2 = 40 m
direction 58 West from North, the angle is measured with respect to the y axis, towards the negative side of the x axis
cos 58 = y₂ / d₂
sin 32 = x₂ / d₂
y₂ = d₂ cos58
x₂ = d₂ sin 58
y₂ = 40 cos 58 = 21.20 m
x₂ = 40 sin 58 = -33.92 m
The negative sign of the x coordinate is due to the fact that the displacement is towards the West (negative of the x axis)
third trajectory
distance d₃ = 39 m
West direction, negative part of x-axis
x₃ = - 39 m
the total distance is
= x₁ + x₂ + x₃
X_{total}l = 14.30 - 33.92 -39
X_{total}X_total = -58.6 m
Y_{total} = y₁ + y₂
Y_{total}= 22.90 + 21.20
Y_{total} = 44.1 m
we give the result in the form of module and angle
we use the Pythagorean theorem
R = [tex]\sqrt{X_{total}^2 + Y_{total}^2 }[/tex]
R =\sqrt58.6^2 + 44.1^2 }
R = 73.34 m
we use trigonometry
tan θ =
θ = tan⁻¹ \frac{ Y_{total} }{ X_{total} }
θ = tan⁻¹ \frac{ 44.1 }{ 58.6 } (44.1 / 58.6)
θ = -37º
this angle is in the second quadrant, given in the form of cardinal coordinates is
θ = 37º North of West
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