Two people decide to find the height of an obelisk. They position themselves 25 feet apart in line with, and on the same side of, the obelisk. If they find that the angles of elevation from the ground where they are standing to the top of the obelisk are 65 degrees and 44 degrees, how tall is the obelisk?
We draw 2 rt. triangles with a common ver. side:
1. Draw a hor. line with points A, B, C.
AB = 25Ft, BC = X Ft.
2. Draw ver. line CD and label it h.
3. Draw AD which makes a 44o angle with
AC.
Draw BD which forms a 65o angle with BC.
Tan65 = h/X
h = X*Tan65 = 2.14X
Tan44 = h/(25+X)
h = (25+X)*Tan44
h = 25.14+0.966X
Replace h with 2.14X
2.14x = 25.14+0.966x
2.14x-0.966x = 25.14
1.174x = 25.14
X = 21.4 Ft.
h = 2.14x = 2.14 * 21.4 = 45.8 Ft.
To find the height of the obelisk, we can use the concept of trigonometry.
Let's denote the height of the obelisk as 'h'.
Since the two people are positioned 25 feet apart, we can consider a right triangle formed by one person, the obelisk, and the vertical line connecting the two people.
Now, let's consider the person who observes the obelisk at an angle of elevation of 65 degrees. In this triangle, the opposite side is the height of the obelisk (h), and the adjacent side is the 25 feet between the two people.
Using the tangent function, we have:
tan(65 degrees) = h / 25
Rearranging the equation, we get:
h = 25 * tan(65 degrees)
Now, let's consider the person who observes the obelisk at an angle of elevation of 44 degrees. In this triangle, the opposite side is again the height of the obelisk (h), and the adjacent side is also the 25 feet between the two people.
Using the tangent function, we have:
tan(44 degrees) = h / 25
Rearranging the equation, we get:
h = 25 * tan(44 degrees)
By plugging in the values for the tangent of 65 degrees and tangent of 44 degrees, we can calculate the value of 'h':
To find the height of the obelisk, we can use the concept of trigonometry, specifically the tangent function.
Let's label the height of the obelisk as 'h.'
From the given information, we have two right triangles formed: one between the first person, the top of the obelisk, and a point on the ground, and another between the second person, the top of the obelisk, and a different point on the ground.
In the first right triangle, the angle of elevation is 65 degrees. This means that the tangent of that angle, tan(65°), is equal to the height of the obelisk divided by the distance between the first person and the obelisk (25 feet). So, we can write the equation as:
tan(65°) = h / 25
Similarly, in the second right triangle, the angle of elevation is 44 degrees. Thus, we can write the equation:
tan(44°) = h / 25
Now, we can solve this system of equations simultaneously to find the height of the obelisk.
First, let's find the value of 'h' using the first equation:
h = tan(65°) * 25
≈ 1.8807 * 25
≈ 47.0175 feet
Next, let's use the second equation to recheck the value of 'h':
h = tan(44°) * 25
≈ 0.9669 * 25
≈ 24.1726 feet
Since we obtained two different values for 'h,' let's take the average of these two values as our final answer:
(h1 + h2) / 2 = (47.0175 + 24.1726) / 2
≈ 71.1901 / 2
≈ 35.595 feet
Therefore, the height of the obelisk is approximately 35.595 feet.