The Hall of Fame exhibits are displayed in an eight-story, 162-foot tower. Pei originally designed a 200-foot tower, but had to reduce its height to meet the requirements of a nearby airport. From the top of the existing tower, an observer sights the entrance to the museum’s plaza with an angle of depression of 18º. What would be the angle of depression to the entrance of the plaza from Pei’s original tower? Round to the nearest tenth of a degree.
X = 162 / tan18 = 499Ft = Dist. from tower to entrance of plaza.
Original Tower:
tanA = Y/X = 200 / 499 = 0.401,
A = 21.8 Deg.
21.9
To find the angle of depression to the entrance of the plaza from Pei's original tower, we need to use the concept of similar triangles.
First, let's define some variables:
Let h be the height of the existing tower (162 feet).
Let x be the distance from the top of the existing tower to the entrance of the museum's plaza.
Let a be the angle of depression from the top of the existing tower to the entrance of the plaza.
Now, we can form two similar right triangles:
1. The first triangle is formed by the existing tower, the distance from the top of the tower to the entrance of the plaza, and the line of sight to the entrance of the plaza. The angle between the line of sight and the horizontal ground is the angle of depression, a.
2. The second triangle is formed by Pei's original tower (200 feet), the unknown distance from the top of the tower to the entrance of the plaza, and the line of sight to the entrance of the plaza. The angle between the line of sight and the horizontal ground is the angle of depression we're trying to find.
Since the two triangles are similar, we can set up the following proportion:
h/x = 200/x'
Here, x' represents the distance from the top of Pei's original tower to the entrance of the plaza (what we want to find).
Now, let's solve for x':
200/x' = 162/x
Cross-multiplying this equation gives us:
200x = 162x'
Divide both sides of the equation by 162:
x' = (200x) / 162
To find the angle of depression from Pei's original tower, we need to find:
arctan(x' / 200)
Substituting the value of x' we found earlier:
arctan((200x) / (162 * 200))
Simplifying the expression:
arctan(x / 162)
Now, we can substitute the known value of the angle of depression from the existing tower (18 degrees) to find the requested angle of depression from Pei's original tower:
18 = arctan(x / 162)
To solve for x, we can take the tangent of both sides:
tan(18) = x / 162
Now, let's calculate the value of x:
x = tan(18) * 162
Finally, we can calculate the angle of depression from Pei's original tower:
arctan(x / 200)
Substituting the value of x we calculated:
arctan((tan(18) * 162) / 200)
To find the answer to the nearest tenth of a degree, calculate the value of this expression. The resulting angle is the angle of depression from Pei's original tower to the entrance of the museum's plaza.