Tamara is flying in a hot air balloon. The hot air balloon is flying at an altitude of 1100 feet. The ground distance to the landing site is 1400 feet.
What is Tamara's angle of depression to the landing site?
What is the actual distance between the hot air balloon and the landing site?
To find the angle of depression, we need to understand the concept of right triangles and trigonometry. In this case, let's consider the hot air balloon's altitude as the height of a right triangle, and the ground distance as the base of the right triangle.
To find Tamara's angle of depression, we can use the tangent function because tangent is the ratio of the opposite side (altitude) to the adjacent side (ground distance).
Step 1: Find the tangent of the angle of depression.
Let's denote the angle of depression as θ.
Tan(θ) = Altitude / Ground Distance
Given: Altitude = 1100 feet, Ground Distance = 1400 feet
Tan(θ) = 1100 / 1400
Step 2: Calculate the angle of depression.
To find the angle of depression, we need to take the inverse tangent (or arctan) of the value obtained in step 1.
θ = arctan(1100 / 1400)
Now we can calculate the actual distance between the hot air balloon and the landing site. To do this, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the altitude is the perpendicular side, the ground distance is the base, and the actual distance is the hypotenuse.
Step 3: Calculate the actual distance.
Using the Pythagorean theorem:
Actual Distance^2 = Altitude^2 + Ground Distance^2
Given: Altitude = 1100 feet, Ground Distance = 1400 feet
Actual Distance^2 = 1100^2 + 1400^2
Thus, the actual distance between the hot air balloon and the landing site is the square root of this value:
Actual Distance = √(1100^2 + 1400^2)