a 60 meter high building is erected vertically on a hillside. An electric post stands vertically on the hill. An observer at the top the building, find the angle of depressions of the top and the bottom of the electric post to be 30 degrees and 42 degrees respectively. if the hill makes an angle of 20 degrees with the horizontal, how high is the electric post.
is the building uphill or downhill from the post?
If the building is downhill from the post, at a horizontal distance of x, then let
a = vertical distance from the base of the building to the base of the post
x = horizontal distance between building and post
h = height of post
a/x = tan20°
(60-a)/x = tan42°
(60-a-h)/x = tan30°
Nope, that gives a negative post height. So, how about the building is uphill from the post? Then we have
a/x = tan20°
(60+a)/x = tan42°
(60+a-h)/x = tan30°
thanks dude, i get the answer..
To solve this problem, we can use the concept of trigonometry. Let's break it down step-by-step:
Step 1: Draw a diagram to visualize the problem.
We have a hillside, where a 60-meter high building is erected vertically. On the hillside, there is also an electric post standing vertically. The observer at the top of the building finds the angle of depression to the top and bottom of the electric post to be 30 degrees and 42 degrees, respectively. The hillside makes an angle of 20 degrees with the horizontal.
Step 2: Label the given information on the diagram.
Let's label the height of the building as 'B' and the height of the electric post as 'P.' We also know the angle of depression from the top of the building to the top of the electric post is 30 degrees, and the angle of depression from the top of the building to the bottom of the electric post is 42 degrees.
Step 3: Identify the right triangles and relevant trigonometric ratios.
In the given problem, we have two right triangles formed. One triangle is formed by the observer at the top of the building, the top of the electric post, and the foot of the electric post. The other triangle is formed by the top of the building, the bottom of the electric post, and the foot of the hillside.
For the first triangle (observer, top of the electric post, and foot of the electric post), we can use the tangent function:
tan(30 degrees) = P / B
For the second triangle (top of the building, bottom of the electric post, and foot of the hill), we can use the tangent function again:
tan(42 degrees) = P / (B + 60 meters)
Step 4: Solve the equations.
From equation 1, we can rewrite it as P = B * tan(30 degrees).
Substitute this value in equation 2:
tan(42 degrees) = (B * tan(30 degrees)) / (B + 60 meters)
Simplifying the equation:
tan(42 degrees) = (B * √(3)/3) / (B + 60 meters)
Step 5: Solve for B, the height of the building.
Multiply both sides of equation 2 by (B + 60 meters):
(B + 60 meters) * tan(42 degrees) = B * √(3)/3
Expand and solve for B:
B * tan(42 degrees) + 60 * tan(42 degrees) = B * √(3)/3
B * (tan(42 degrees) - √(3)/3) = -60 * tan(42 degrees)
B = (-60 * tan(42 degrees)) / (tan(42 degrees) - √(3)/3)
Calculating the value:
B ≈ 33.96 meters
Step 6: Calculate the height of the electric post.
Using equation 1: P = B * tan(30 degrees)
P = 33.96 meters * √(3)/3
Calculating the value:
P ≈ 19.62 meters
So, the height of the electric post is approximately 19.62 meters.
To find the height of the electric post, we can use the trigonometric concept of angles of depression.
Let's start by visualizing the scenario. We have a building erected vertically on a hillside, and there is an electric post vertically standing on the hill. The observer at the top of the building can see the top and the bottom of the electric post at angles of depression of 30 degrees and 42 degrees, respectively. The hill makes an angle of 20 degrees with the horizontal.
To solve this problem, we will break it down into smaller parts. Let's use the following variables:
h = height of the electric post (what we want to find)
d = horizontal distance between the building and the electric post
x = vertical distance between the top of the building and the electric post
y = vertical distance between the bottom of the building and the electric post
From the given information, we know that the distance (d) between the building and the electric post is the same at both the top and the bottom. So, we can set up the following equation using trigonometry:
tan(30) = x / d
We can rearrange this equation to solve for x:
x = d * tan(30)
Similarly, we can set up another equation using trigonometry for the bottom of the building:
tan(42) = y / d
Rearranging this equation gives us:
y = d * tan(42)
Now, we can consider the hill's angle of 20 degrees. From the given information, we can infer that the height of the electric post (h) is the sum of the vertical distances (x and y) from the top and bottom of the building to the electric post. So we have:
h = x + y
Substituting the values of x and y, we get:
h = (d * tan(30)) + (d * tan(42))
Finally, we can substitute in the given angle of the hill (20 degrees) to solve for h.
h = d * (tan(30) + tan(42))
Now, given that the height of the building is 60 meters, we can set up another equation:
h + 60 = d * tan(20)
Rearranging this equation, we obtain:
d = (h + 60) / tan(20)
Substituting this value for d into the equation for h, we get:
h = [(h + 60) / tan(20)] * (tan(30) + tan(42))
Simplifying this equation will give us the height of the electric post (h).