A pilot in a plane at an altitude of 22,000 feet observes that the angle of depression to a nearby airport is 26 degrees. How many miles is the airport from a point on the ground directly below the plane?
Well, let's do some math, shall we? But don't worry, I promise to keep it light and entertaining!
First, we need to determine the distance from the plane to the airport. Now, since our pilot is such a keen observer, we know that the angle of depression is 26 degrees. So, imagine the plane looking down at the airport with a bit of a sad face, like it's feeling a little down. Poor little plane!
Now, imagine drawing a line from the plane to the point on the ground directly below it, forming a right triangle. The angle of depression is the angle between the hypotenuse (the line from the plane to the airport) and the adjacent side (the line from the plane to the point on the ground). Are you with me so far?
Next, let's use a little bit of trigonometry magic. We can use the tangent function to find the length of the adjacent side. We know that the tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side.
In this case, the opposite side is the altitude of the plane, which is 22,000 feet. The tangent of 26 degrees is approximately 0.4877 (rounded to four decimal places, because I'm fancy like that).
So, let's set up our equation: tangent(26 degrees) = 22,000 feet / length of adjacent side. Solving for the length of the adjacent side, we get:
length of adjacent side = 22,000 feet / tangent(26 degrees)
≈ 22,000 feet / 0.4877
≈ 45,111.17 feet.
But wait, we want the answer in miles, not feet. Silly me, I forgot to convert! There are approximately 5,280 feet in a mile, so let's divide our answer by 5,280:
length of adjacent side in miles = 45,111.17 feet / 5,280
≈ 8.54 miles.
Ta-da! So, the airport is approximately 8.54 miles from a point on the ground directly below the plane. Safe travels, my friend! And remember, always keep your spirits high, even when your plane is feeling a little down.
To solve this problem, we can use trigonometry. We can draw a right triangle, where the angle of depression is 26 degrees.
Let's consider the vertical leg of the triangle as the altitude of the plane, which is 22,000 feet. The horizontal leg represents the distance from the point on the ground directly below the plane to the airport.
Using trigonometry, we can find the length of the horizontal leg of the triangle.
The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the angle is 26 degrees, the opposite side is 22,000 feet (altitude), and we need to find the adjacent side (distance to the airport).
So, we have:
tan(26°) = opposite/adjacent
tan(26°) = 22,000/adjacent
Now, we can solve for the adjacent side (distance to the airport):
adjacent = 22,000 / tan(26°)
adjacent ≈ 45,375.29 feet (rounded to two decimal places)
Since we need the distance in miles, let's convert feet to miles. There are 5,280 feet in a mile.
distance to the airport ≈ 45,375.29 / 5280 ≈ 8.60 miles (rounded to two decimal places)
Therefore, the airport is approximately 8.60 miles away from the point on the ground directly below the plane.
To solve this problem, we can use trigonometry. We can assume the plane is directly above point A on the ground and the airport is directly below point B. We need to find the distance AB.
Let's break down the given information:
- The altitude of the plane is 22,000 feet. This means the height of the triangle formed by the plane, the point on the ground directly below it, and the airport is 22,000 feet.
- The angle of depression is 26 degrees. This angle is the angle formed between the horizontal line connecting the plane to the airport and the line connecting the plane to the ground point below it.
To find the distance AB, we can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle.
In this case, the opposite side is the altitude (22,000 feet), and the adjacent side is the distance we want to find (AB). We can set up the equation as follows:
tangent(26 degrees) = opposite / adjacent
tan(26°) = 22,000 / AB
Now, we can solve for AB by rearranging the equation:
AB = 22,000 / tan(26°)
Using a scientific calculator, we can find the value of tangent(26°) to be approximately 0.4877. So, the equation becomes:
AB = 22,000 / 0.4877
Evaluating this expression, we find that AB is approximately 45,092.4 feet.
To convert this distance into miles, we need to divide by the number of feet in a mile (5,280 feet). So, the final answer is:
AB ≈ 45,092.4 / 5,280 ≈ 8.53 miles
Therefore, the airport is approximately 8.53 miles away from the point on the ground directly below the plane.
tan 26 = (22,000/d)
d tan 26 = 22,000
d = 22,000 / tan 26 = 45,107