An airplane flies 400 km west from city A to city B, then 200km northwest to city c and finally 100 km north to city D. How far it is from city a to d?
Draw the picture. You will need to resolve vector B into its N and W components (200cos45, 200sin45).
To find the total distance from City A to City D, we need to calculate the distance traveled in each leg of the journey and add them up.
1. Distance from City A to City B: The airplane flies 400 km west.
2. Distance from City B to City C: The airplane flies 200 km northwest.
3. Distance from City C to City D: The airplane flies 100 km north.
To determine the distance from City A to City D, we need to find the diagonal distance of the triangle formed by City A, City B, and City D.
Using the Pythagorean theorem (a^2 + b^2 = c^2), where a and b are the legs of the right triangle and c is the hypotenuse, we can find the distance from City A to City D:
a = 400 km (distance from City A to City B)
b = 100 km (distance from City C to City D)
(a^2) + (b^2) = (c^2)
(400^2) + (100^2) = (c^2)
160,000 + 10,000 = c^2
170,000 = c^2
Taking the square root of both sides, we find:
c = √170,000
Calculating this, we get:
c ≈ 412.3 km
Therefore, the distance from City A to City D is approximately 412.3 km.
To find the total distance from city A to city D, we need to calculate the combined distances of each leg of the journey.
First, we'll calculate the distance from city A to city B, which is 400 km.
Next, we'll calculate the distance from city B to city C. The airplane travels 200 km northwest, which implies a change in both latitude and longitude. To calculate this distance, we can use the Pythagorean theorem. We have a right-angled triangle with sides of 200 km and 200 km. Using the formula c^2 = a^2 + b^2, where c is the hypotenuse (the distance between city B and city C) and a and b are the other two sides, we can solve for c:
c^2 = 200^2 + 200^2
c^2 = 40000 + 40000
c^2 = 80000
c ≈ 282.84 km (rounded to two decimal places)
Finally, we'll calculate the distance from city C to city D, which is 100 km north.
To find the total distance from city A to city D, we add up the distances:
400 km + 282.84 km + 100 km = 782.84 km (rounded to two decimal places)
Therefore, the airplane is approximately 782.84 km away from city A when it reaches city D.