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
24
∘
and that the angle of depression to the bottom of the tower is
21
∘
. How tall is the tower?
450 tan(24º) + 450 tan(21º)
Tan21 = 450/x
X = 1172 Ft. = hor. distance between bldg. and tower.
Tan24 = h1/1172
h1 = 1172*Tan24 = 522 Ft.
ht. = 522 + 450 =
To find the height of the tower, we can use trigonometric ratios.
Let's assume the height of the tower is h.
From the window in the building, the person is determining the angle of elevation to the top of the tower, which is 24°. This means that if we draw a right triangle with the height of the tower as the vertical side and the distance between the tower and the building as the horizontal side, the angle between them is 24°.
Similarly, the angle of depression to the bottom of the tower is 21°. This means the angle between the horizontal side (distance between the tower and the building) and the vertical side (height of the tower) is 21°.
Now, we can use the tangent function to find the height of the tower:
tan(24°) = h / 450
Rearranging the equation, we get:
h = 450 * tan(24°)
Using a calculator, we find:
h ≈ 191.1 feet
Therefore, the tower is approximately 191.1 feet tall.
To find the height of the tower, we need to use trigonometric functions based on the given angles of elevation and depression.
Let's consider the triangle formed by the person, the top of the tower, and the bottom of the tower. The distance from the person to the tower is the base of the triangle, and the height of the tower is the height of the triangle.
1. Let's start by finding the height of the triangle using the angle of depression. The angle of depression is the angle between a horizontal line and the line of sight from the person to the bottom of the tower. In this case, it is 21∘.
We can use the tangent function to find the height of the triangle. The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.
tangent(21∘) = height of the triangle / 450 feet
Rearranging the equation, we get:
height of the triangle = tangent(21∘) * 450 feet
2. Next, let's find the length of the base of the triangle, which is the distance from the person to the tower. In this case, it is given as 450 feet.
3. Finally, we can find the height of the tower using the angle of elevation. The angle of elevation is the angle between a horizontal line and the line of sight from the person to the top of the tower. In this case, it is 24∘.
We can use the tangent function again to find the height of the tower. Using the same reasoning as before:
tangent(24∘) = height of the triangle / 450 feet
Rearranging the equation, we get:
height of the triangle = tangent(24∘) * 450 feet
Now that we have the height of the triangle calculated using both angles, we can equate them since they represent the same height:
tangent(21∘) * 450 feet = tangent(24∘) * 450 feet
Canceling out the 450 feet on both sides, we get:
tangent(21∘) = tangent(24∘)
Now we can solve for the height of the tower by finding the value of the tangent of either of the angles. In this case, we'll use the angle of elevation (24∘):
height of the tower = tangent(24∘) * 450 feet
Using a scientific calculator, we can calculate the tangent of 24∘, which is approximately 0.445
height of the tower = 0.445 * 450 feet
Calculating the height, we get:
height of the tower ≈ 200.25 feet
Therefore, the tower is approximately 200.25 feet tall.