Question

Two flights leave from the same airport. Determine which flight is farther from the airport.

Flight 1: Flies 210 miles due south, then turns 70° toward west and flies 80 miles.
Flight 2: Flies 80 miles due north, then turns 50° toward east and flies 210 miles.

Answer 1

To determine which flight is farther, we calculate the resultant displacements for both flights using vector addition and compare them. This involves the Law of Cosines and trigonometry to find the combined effects of the two flight segments in each case.

Step-by-step explanation:

To determine which flight is farther from the airport, we can use vector addition and trigonometry to find the resultant displacement from the starting point for both flights. For this comparison, we will use Flight 1 and Flight 2 as examples.

Flight 1 Calculation:

Flight 1 flies 210 miles due south.

Then, it turns 70° toward west and flies 80 miles.

To find the resultant displacement, we use the Law of Cosines or vector addition.
Resultant Displacement = √(210^2 + 80^2 - 2*210*80*cos(110°))

Flight 2 Calculation:

Flight 2 flies 80 miles due north.

It then turns 50° toward east and flies 210 miles.

Similarly, to find the resultant displacement for Flight 2, we also use the Law of Cosines or vector addition.
Resultant Displacement = √(80^2 + 210^2 - 2*80*210*cos(130°))

By calculating the resultant displacements for each flight, we can compare the distances and determine which flight is farther from the airport.

Answer 2

Flight 1 is farther from the airport.

Step-by-step explanation:

To determine which flight is farther from the airport, we need to break down the two flights into their component distances.

Flight 1: Flies 210 miles due south and then turns 70° toward the west and flies 80 miles.

Using trigonometry, we can find the south and west components of Flight 1.

The south component is 210 * sin(70°) = 197.29 miles, and the west component is:

210 * cos(70°) = 71.93 miles.

Flight 2: Flies 80 miles due north and then turns 50° toward the east and flies 210 miles.

Again using trigonometry, we can find the north and east components of Flight 2.

The north component is 80 * sin(50°) = 61.39 miles, and the east component is 80 * cos(50°) = 51.91 miles.

Now we can calculate the distances of Flight 1 and Flight 2 from the airport.

The distance from the airport for Flight 1 is the magnitude of the south and west components:

sqrt((197.29)^2 + (71.93)^2)

≈ 209.99 miles.

The distance from the airport for Flight 2 is the magnitude of the north and east components:

sqrt((61.39)^2 + (51.91)^2)

≈ 81.47 miles.

Therefore, Flight 1 is farther from the airport.