an airplane and a helicopter leave an airport in montana at the same time.The aircrafts velocities:v=225mph@20 degree,and v=102mph@49 degree.If these velocities are maintained which craft will cross the borderinto canada first? or do they arrive at the same time. Try the suggestions below or type a new qu

helicopter

To determine which aircraft will cross the border into Canada first, we need to calculate their respective arrival times based on their velocities.

First, let's convert the velocities from polar coordinates to Cartesian coordinates, so we can work with x and y components:

For the airplane's velocity:
v_airplane = 225 mph at 20 degrees
v_airplane_x = 225 * cos(20) = 210.9403 mph (approx.)
v_airplane_y = 225 * sin(20) = 77.2295 mph (approx.)

For the helicopter's velocity:
v_helicopter = 102 mph at 49 degrees
v_helicopter_x = 102 * cos(49) = 65.1813 mph (approx.)
v_helicopter_y = 102 * sin(49) = 77.0348 mph (approx.)

Now, let's calculate the time it takes for each aircraft to travel from the airport to the Canadian border. We'll assume a straight-line distance from the airport to the border for simplicity.

Let's say the distance from the airport to the border is 150 miles (as an example).

For the airplane:
time_airplane = distance / v_airplane = 150 / 210.9403 ≈ 0.7116 hours or 42.696 minutes

For the helicopter:
time_helicopter = distance / v_helicopter = 150 / 65.1813 ≈ 2.2995 hours or 137.969 minutes

Comparing the arrival times, we can see that the airplane will cross the border into Canada first, arriving approximately 42.696 minutes after leaving the airport. The helicopter will arrive later, approximately 137.969 minutes (about 2 hours and 18 minutes) after leaving the airport.

Therefore, the airplane will arrive in Canada before the helicopter.

To determine which aircraft will cross the border into Canada first, we need to calculate the time it takes for each aircraft to reach the border. We can use the following steps:

1. Convert the velocities given for each aircraft to their respective horizontal velocities. To do this, we can use the formula: horizontal velocity = velocity * cos(angle).

For the airplane:
Horizontal velocity = 225 mph * cos(20 degrees) ≈ 213.44 mph

For the helicopter:
Horizontal velocity = 102 mph * cos(49 degrees) ≈ 65.84 mph

2. Now we can calculate the time it takes for each aircraft to reach the border, assuming they maintain their velocities.

Let's assume the distance from the airport to the border is "d" miles.

For the airplane:
Time taken = distance / velocity = d / 213.44 mph

For the helicopter:
Time taken = distance / velocity = d / 65.84 mph

3. Simplifying the equations, we have:

Time taken for the airplane = d / 213.44 mph
Time taken for the helicopter = d / 65.84 mph

Since both aircrafts leave the airport at the same time, we can ignore the "d" distance factor in both equations.

Therefore, we can conclude that the aircraft which reaches the border first is the one with the shortest time taken. Let's calculate the time ratio:

Time ratio = (Time taken for the airplane) / (Time taken for the helicopter)
= (d / 213.44 mph) / (d / 65.84 mph)
= (65.84 mph) / (213.44 mph)

Now let's compare the time ratio to determine which aircraft will reach the border first:

If (65.84 mph) / (213.44 mph) ≈ 0.308, it means that the helicopter takes approximately 0.308 times as long as the airplane.

Therefore, the helicopter will reach the Canadian border first, as it takes less time to cover the same distance compared to the airplane.

It's important to note that this analysis assumes both aircraft maintain their velocities and travel in a straight line towards the border without accounting for winds, air traffic, or other external factors that may affect their travel time.