An airplane flies 200 km due west from city A to city B and then 270 km in the direction of 31.0° north of west from city B to city C.
(a) In straight-line distance, how far is city C from city A?
(b) Relative to city A, in what direction is city C?
AC = 200 km @ 180o + 270 km @ 149o.
X=Hor.=200*cos180 + 270*cos149=-431.4 km
Y=Ver.=200*sin180 + 270*sin149=139.1 km.
a. (AC)^2 = X^2 + Y^2 = 205,454.77
AC = 453.3 km.
b. tanA = Y/X = 139.1/431.4 = 0.32244
A = 17.9o North of West.
To solve this problem, we can break down the airplane's flight into its North-South and East-West components using trigonometry. Let's find the answers step by step.
(a) To calculate the straight-line distance between city A and city C, we can use the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) of a right triangle is equal to the sum of the squares of the other two sides.
First, let's calculate the North-South component of the airplane's flight:
Distance north of west = 270 km * sin(31°)
Distance north of west ≈ 139.46 km (rounded to two decimal places)
Now, let's calculate the East-West component of the airplane's flight:
Distance west = 200 km
Using these values, we can apply the Pythagorean theorem to find the straight-line distance between city A and city C:
Distance AC = √((Distance west)^2 + (Distance north of west)^2)
Distance AC = √((200 km)^2 + (139.46 km)^2)
Distance AC ≈ √(40000 km^2 + 19423.4416 km^2)
Distance AC ≈ √(59423.4416 km^2)
Distance AC ≈ 243.58 km (rounded to two decimal places)
Therefore, the straight-line distance between city A and city C is approximately 243.58 km.
(b) To determine the direction of city C relative to city A, we can use trigonometric ratios. In this case, we need to find the angle that the line from city A to city C makes with the west direction.
Using the East-West and North-South components calculated earlier:
Distance west = 200 km
Distance north of west = 139.46 km
To find the angle, we can use the tangent function:
tan(angle) = (Distance north of west) / (Distance west)
tan(angle) = 139.46 km / 200 km
angle = arctan(139.46 km / 200 km)
angle ≈ 37.80° (rounded to two decimal places)
Therefore, relative to city A, city C is approximately 37.80° north of west.