A man standing 9 meters above the ground observes the angles of elevation and depression of the top and bottom,of the top of the monument in luneta 6 degrees and 50 minutes and 7 degrees and 30 minutes respectively.Find the height of the monument?

did you make a diagram ?

let the top of the monument be A and the bottom be B
let the person's position be at P
Draw a horizontal from P to AB to meet AB at C.
then CB = 9
angle APC = 6°' 50' or 6.8333°
angle BPC = 7° 30' or 7.5°

Using triangle PBC you can find PC using tan7.5°
Once you have PC you can go into the other triangle and again using the tangent ratio find AC

Height of monument = AC + 9

13

To find the height of the monument, we can use trigonometric ratios and set up the following equations:

Let the height of the monument be "h" meters.

1. From the man's perspective, the angle of elevation from the bottom of the monument to the top is 6°50'. This forms a right-angled triangle with the height of the monument, the distance from the man to the bottom of the monument (9m), and a line connecting the man's eyes to the top of the monument.
Using the tangent function:
tan(6°50') = h / 9

2. From the man's perspective, the angle of depression from the top of the monument to the bottom is 7°30'. This also forms a right-angled triangle with the height of the monument, the distance from the man to the bottom of the monument (9m), and a line connecting the man's eyes to the top of the monument.
Using the tangent function:
tan(7°30') = h / 9

Now, we can solve these equations to find the height of the monument.

Step 1: Convert the angles from degrees and minutes to decimal form.
6°50' = 6 + (50/60) = 6.8333°
7°30' = 7 + (30/60) = 7.5°

Step 2: Calculate the height of the monument using the tangent function.
tan(6.8333°) = h / 9
h = 9 * tan(6.8333°)
h ≈ 9 * 0.1196
h ≈ 1.0764

tan(7.5°) = h / 9
h = 9 * tan(7.5°)
h ≈ 9 * 0.1316
h ≈ 1.1844

The height of the monument is approximately 1.18 meters.

To find the height of the monument, we can use the concept of trigonometry and the given angles of elevation and depression.

Let's denote the height of the monument as 'h'.

From the given information, we know that the man is standing 9 meters above the ground. Therefore, the height of his eye level is 9 meters.

Now, let's consider the angle of elevation. The angle of elevation is the angle formed between the horizontal line and the line of sight from the observer's eye to the top of the monument. In this case, the angle of elevation is 6 degrees and 50 minutes.

Similarly, let's consider the angle of depression. The angle of depression is the angle formed between the horizontal line and the line of sight from the observer's eye to the bottom of the monument. In this case, the angle of depression is 7 degrees and 30 minutes.

Using trigonometry, we can set up the equations:

(1) tan(angle of elevation) = height of monument / distance from observer to monument
(2) tan(angle of depression) = (height of monument - height of observer's eye) / distance from observer to monument

Now, let's convert the angles from degrees and minutes to decimal form for easier calculations:

Angle of elevation = 6 degrees + (50/60) degrees = 6.83 degrees
Angle of depression = 7 degrees + (30/60) degrees = 7.5 degrees

Substituting the known values into the equations, we get:

(1) tan(6.83 degrees) = h / x
(2) tan(7.5 degrees) = (h - 9) / x

Here, 'x' represents the distance from the observer to the monument. Since the observer is standing at ground level, we can assume that x is the same for both equations.

Now, we can solve these equations simultaneously to find the value of h. Rearranging the equations, we get:

(1) x = h / tan(6.83 degrees)
(2) x = (h - 9) / tan(7.5 degrees)

Since x is the same in both equations, we can equate them:

h / tan(6.83 degrees) = (h - 9) / tan(7.5 degrees)

Cross multiplying and simplifying, we have:

h * tan(7.5 degrees) = (h - 9) * tan(6.83 degrees)

Expanding this equation, we get:

h * 0.1316 = h * 0.1251 - 9 * 0.1251

Simplifying further:

0.1316h = 0.1251h - 1.1259

Rearranging and combining like terms:

0.0065h = 1.1259

Dividing both sides by 0.0065:

h = 1.1259 / 0.0065

Therefore, the height of the monument is approximately h = 173.22 meters.