A plane is 11.9 miles off the ground as it makes its descent to land at the Metropolitan airport (46 miles away). The pilot aims the plane downward at a 12-degree angle. To the nearest mile, what distance will the plane have to travel in the air to land at the airport?
11.9/d = sin 12°
To find the distance the plane will have to travel in the air to land at the airport, we can use trigonometry.
First, let's visualize the scenario: the plane is descending at a 12-degree angle from a height of 11.9 miles to land at the Metropolitan airport, which is 46 miles away horizontally.
Now, we can split the motion of the plane into two components: horizontal and vertical. The horizontal component represents the distance the plane will travel in the air, and the vertical component represents the change in height.
To find the distance traveled horizontally, we need to calculate the length of the adjacent side of the angle triangle formed by the plane's motion. We can use the trigonometric function cosine to find this value.
cos(angle) = adjacent / hypotenuse
The angle in this case is 12 degrees, the adjacent side is the horizontal distance we are looking for, and the hypotenuse is the total distance the plane will have to travel from its starting position to the airport, which is 46 miles in this case.
cos(12 degrees) = adjacent / 46 miles
Next, we can rearrange the equation to solve for the adjacent side (horizontal distance):
adjacent = cosine(12 degrees) * 46 miles
Using a scientific calculator or an online calculator, we can find the cosine of 12 degrees, which is approximately 0.9781.
adjacent = 0.9781 * 46 miles
Multiplying these values together, we get:
adjacent ≈ 44.9366 miles
Now, let's round this value to the nearest mile, which gives us:
horizontal distance ≈ 45 miles
Therefore, the plane will have to travel approximately 45 miles in the air to land at the Metropolitan airport.