From a point A is 8.20 meters above level ground,the angle of elevation of the top of a building is 31degrees 20 min and the angle of depression of the base of the building is 12degree 50 min. Approximate the heighth of the building
If the distance of A is d, and the building's height is h,
(h-8.2)/d = tan 31°20'
8.2/d = tan 12°50'
so, equating d,
(h-8.2)/tan 31°20' = 8.2/tan 12°50'
(h-8.2)/.6088 = 8.2/.2278
h = 30.11 m
Well, if you're looking for an approximate height of the building, I'd be happy to help with a dash of humor!
First, let's focus on the angle of elevation. Imagine the angle saying, "Hey, building, I'm going to look up at you with awe!" That's essentially what it means—we're looking up from point A to the top of the building.
Now, let's consider the angle of depression. This angle says, "Oh, building, I'm going to look down at your base." It's like hugging the building and looking down, but don't get too close, or you might trip!
Alright, let's crunch some numbers. We have an angle of elevation of 31 degrees, 20 minutes and an angle of depression of 12 degrees, 50 minutes.
To calculate the height of the building, we'll use some trigonometry. We can start by finding the length of side AB, which is the height of the building. We'll call this length h.
Using the tangent function, we can relate the angle of elevation to the height of the building:
tan(31° 20') = h / 8.20
Similarly, we can use the tangent function with the angle of depression to relate the height of the building to the distance BC (the base of the building):
tan(12° 50') = h / BC
Now, let's solve these equations to find an approximation of the height of the building. Just remember, to calculate accurately, use the actual degrees and minutes, not the silly ones I mentioned earlier!
To approximate the height of the building, we can use trigonometry.
Step 1: Convert the angles from degrees and minutes to decimal form.
- Angle of elevation = 31 degrees + 20/60 = 31.33 degrees
- Angle of depression = 12 degrees + 50/60 = 12.83 degrees
Step 2: Set up a right triangle with the known information.
- Let H be the height of the building.
- The vertical side of the triangle represents the height above the ground (8.20 meters).
- The angle of elevation (31.33 degrees) can be used to find the opposite side length (H).
- The angle of depression (12.83 degrees) can be used to find the adjacent side length (D), which represents the horizontal distance from point A to the building.
Step 3: Use trigonometry to find the missing side lengths.
- Use the tangent function to find H: tan(31.33 degrees) = H / 8.20 meters
- Rearrange the equation: H = 8.20 meters * tan(31.33 degrees)
Step 4: Calculate the height of the building.
- Use a calculator to find the value of tan(31.33 degrees) and multiply it by 8.20 meters.
- H ≈ 8.20 meters * 0.618 ≈ 5.07 meters
Therefore, the approximate height of the building is 5.07 meters.
To approximate the height of the building, we can use trigonometric concepts. Let's break down the problem step by step:
1. Draw a diagram: Draw a right-angled triangle to represent the situation. Label the point A as the observer's position, the top of the building as point B, and the base of the building as point C. The vertical line between points A and B represents the height of the building.
2. Determine the given angle of elevation: The angle of elevation is the angle formed between the ground and the line of sight from the observer to the top of the building. In this case, the angle of elevation is 31 degrees 20 minutes or 31° 20'.
3. Determine the given angle of depression: The angle of depression is the angle formed between the horizontal ground and the line of sight from the observer to the base of the building. In this case, the angle of depression is 12 degrees 50 minutes or 12° 50'.
4. Find the complementary angles: Since the angle of elevation and the angle of depression are formed by the same line of sight, their sum will be 90 degrees (as they are complementary angles).
Angle of elevation + Angle of depression = 90°
Convert both angles to decimal form for easier addition:
Angle of elevation: 31° 20' = 31 + (20/60) = 31.33 degrees
Angle of depression: 12° 50' = 12 + (50/60) = 12.83 degrees
Angle of elevation + Angle of depression = 31.33° + 12.83° = 44.16° (approximately)
5. Use trigonometric ratios: In this triangle, we can use the tangent ratio to find the height of the building.
tan(angle) = Opposite / Adjacent
Since we want to find the height of the building (the opposite side), and we know the angle of elevation (44.16 degrees) and the distance from the observer to the building base (8.20 meters), we set up the equation as follows:
tan(44.16°) = Height / 8.20
6. Solve for the height: Rearrange the equation to isolate the height:
Height = tan(44.16°) * 8.20
Using a scientific calculator, evaluate the right side of the equation:
Height ≈ 8.20 * tan(44.16°) ≈ 8.20 * 0.9929 ≈ 8.13 meters
Therefore, the approximate height of the building is 8.13 meters.