John walks 1.0 km north, turns and walks 1.2 km at 50 degrees north of due east, and then turns and walks .8 km at 25 degrees north of east. what is the resultant distance and direction he has flown
D = 1km[90o] + 1.2km[50o] + 0.8km[25o]
X=1*cos90 + 1.2cos50 + 0.8*cos25=1.50 km
Y=1*sin90+1.2*sin50+0.8*sin25=2.257 km
tanA = Y/X = 2.257/1.50 = 1.50892
A = 56.5o = Direction.
D = Y/sinA = 2.257/sin56.5=2.71m[56.5o].
To find the resultant distance and direction, we can break down John's movement into its north and east components, and then sum them up.
Step 1: Calculate the north and east components for each leg of the journey.
For the first leg (1.0 km north), the north component is 1.0 km (since he only walks north) and the east component is 0 km.
For the second leg (1.2 km at 50 degrees north of due east), we need to calculate the north and east components using trigonometry. We can use the following equations:
north component = distance * cos(angle)
east component = distance * sin(angle)
north component = 1.2 km * cos(50°) ≈ 0.769 km
east component = 1.2 km * sin(50°) ≈ 0.921 km
For the third leg (0.8 km at 25 degrees north of east), we can use the same equations:
north component = 0.8 km * cos(25°) ≈ 0.722 km
east component = 0.8 km * sin(25°) ≈ 0.342 km
Step 2: Add up the north and east components.
Total north component = 1.0 km + 0.769 km + 0.722 km ≈ 2.491 km
Total east component = 0 km + 0.921 km + 0.342 km ≈ 1.263 km
Step 3: Use the Pythagorean theorem to find the resultant distance.
resultant distance = √(north component^2 + east component^2)
resultant distance = √(2.491 km^2 + 1.263 km^2) ≈ √(6.203481 km^2) ≈ 2.490 km
Step 4: Use trigonometry to find the direction.
direction = atan(east component / north component)
direction = atan(1.263 km / 2.491 km) ≈ atan(0.507) ≈ 26.6°
Therefore, the resultant distance John has covered is approximately 2.490 km in a direction of 26.6 degrees north of due east.