A sailboat travels 13 miles at 27 degrees east of north. Then travels 18 miles at 35 degrees south of east. What is the result of the entire trip? (Distance and direction)
It would be also be great if you told me how you worked it out. Thanks!
Step 1: make a diagram
on mine, the trip forms a triangle with sides of 13 and 18 miles and an angle of 82°
between them
a clear case of the cosine law: let the distance be d
d^2 = 13^2 + 18^2 - 2(13)(18)cos82°
continue, let me know what you get
Wouldn't the angle be 62° rather than 82°?
Because 27+35=62?
All angles are measured CW from +y-axis.
D = 13mi[27o] + 18mi[125o]
D = (13*sin27+18*sin125) + (13*cos27+18*cos125)I
D = 20.6 + 1.26i = 20.64mi[86.5o] CW.
To find the result of the entire trip, we can use vector addition. We can break down each leg of the trip into its x and y components.
The first leg of the trip is 13 miles at 27 degrees east of north. To find the x and y components of this leg, we can use trigonometry:
x1 = 13 * sin(27)
y1 = 13 * cos(27)
The second leg of the trip is 18 miles at 35 degrees south of east. Again, we can use trigonometry to find the x and y components:
x2 = 18 * cos(35)
y2 = -18 * sin(35) (we use -sin(35) since it's south of east)
Now, we can add up the x and y components of the two legs to find the total displacement:
x_total = x1 + x2
y_total = y1 + y2
Using these values, we can find the magnitude (distance) and direction of the total displacement using the Pythagorean theorem and inverse trig functions:
distance = sqrt(x_total^2 + y_total^2)
direction = atan2(x_total, y_total)
Performing the calculations, the distance of the entire trip is approximately 21.83 miles, and the direction is approximately -40 degrees (taking north as 0 degrees and going clockwise as positive).
Therefore, the result of the entire trip is a distance of 21.83 miles and a direction approximately 40 degrees south of west.