A radio tower is located 450 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is 29 degrees and that the angle of depression to the bottom of the tower is 22 degrees. How tall is the tower?
the height h is
h = 450tan29° + 450tan22° = 431
thank you!!
Well, isn't it just like a tower to be so high-maintenance and make people crane their necks at all sorts of angles? Anyway, let's get to cracking this case.
First, let's find the distance from the top of the tower to the window. We can imagine drawing a line from the person's eyes to the top of the tower, forming a right triangle. The angle of elevation is 29 degrees, opposite to the height of the tower, and the adjacent side is the distance from the tower to the window.
Using a bit of trigonometry (because, you know, it's always trigonometry's time to shine), we can say that tan(29 degrees) = opposite/adjacent. The opposite, in this case, is the height of the tower, let's call it "h", and the adjacent is the distance from the tower to the window, 450 feet.
So, tan(29 degrees) = h/450. Solving for h gives us h = 450 * tan(29 degrees).
Now, let's calculate the distance from the bottom of the tower to the window. We can apply the same logic but use the angle of depression instead. Once again, we have a right triangle, with the angle of depression being 22 degrees, and we're looking for the adjacent side: let's call it "d".
Using the trusty old trigonometry again, we have tan(22 degrees) = h/d. This time, the opposite side is still the height of the tower, "h", and the adjacent side is the distance from the bottom of the tower to the window, "d".
Solving for "d" gives us d = h/tan(22 degrees).
Now, here comes the big reveal! We know that the distance from the top of the tower to the window, "h", is the same as the distance from the bottom of the tower to the window, "d". So we can set these two equations equal to each other:
450 * tan(29 degrees) = h/tan(22 degrees).
Now we can solve for h by taking the product of the two tangent values, and then dividing both sides by tan(22 degrees):
h = (450 * tan(29 degrees)) / tan(22 degrees).
Calculating this equation will give us the height of the tower, and it's not my place to do the dirty work for you. So go on ahead and grab your calculator to find out how tall that tower is. Enjoy!
To find the height of the tower, we can use the concept of trigonometry. Let's break down the problem and identify the relevant information.
Given:
- Distance from the building to the tower: 450 feet
- Angle of elevation to the top of the tower: 29 degrees
- Angle of depression to the bottom of the tower: 22 degrees
First, let's draw a diagram to visualize the situation.
A
/|
/ |
/ | h (height of tower)
building / |
/ |
/ |
/ |
/θ1 |θ2
/_______|
450 ft tower-base
In the diagram, the tower is represented by AB, and the height of the tower is represented by h. The angle of elevation is θ1, and the angle of depression is θ2.
To find the height of the tower (h), we'll use the tangent function, as we have the opposite and adjacent sides of the right triangle.
From the angle of elevation, we have:
tan(θ1) = h / 450 ft
From the angle of depression, we have:
tan(θ2) = h / 450 ft
Now, let's solve for h in both equations.
From the angle of elevation:
h = 450 ft * tan(θ1)
From the angle of depression:
h = 450 ft * tan(θ2)
Using a scientific calculator, plug in the values of θ1 (29 degrees) and θ2 (22 degrees) into the respective equations.
From the angle of elevation:
h = 450 ft * tan(29 degrees) ≈ 225.58 ft
From the angle of depression:
h = 450 ft * tan(22 degrees) ≈ 177.07 ft
Therefore, the height of the tower is approximately 225.58 feet.