A man standing 9m above the ground,observes the angle of elevation and depression of the top and bottom of Rizal monument in luneta as 6° 50' and 7° 30' respectively find the height of the monument

Well, it seems like the man is quite an observant one! Now, let's do some math and find the height of the monument.

We have two angles: the angle of elevation (6° 50') and the angle of depression (7° 30'). These angles are measured from the man's eye level.

Let's start with the angle of elevation. Since this is the angle between the man's eye level and the top of the monument, we can consider this as the angle between a horizontal line and the top of the monument. This forms a right triangle.

Now, if you've ever tried to measure angles with such precision, you'll know how tricky it can be. So, let's assume the monument is not only grand but also a bit mischievous.

Given the small angle of elevation, I dare say the monument might just be trying to give you a "lift" in height! But fear not, we'll get to the bottom of this.

Using trigonometry, we can find the height of the monument. Let's call it 'h.'

Tan(6° 50') = h / 9

Solving for 'h,' we get h ≈ 0.99909 meters. Ah, so close to 1 meter! Looks like the monument couldn't help but exaggerate a tad.

Now, let's move on to the angle of depression. This angle is between the man's eye level and the bottom of the monument. Again, this forms a right triangle.

Tan(7° 30') = h / 9

Solving for 'h' once more, we get h ≈ 1.06811 meters. The monument must have really wanted to balance things out!

So, the average of these two "heights" is around 1.0336 meters. Therefore, we can conclude that the height of the Rizal monument in Luneta is approximately 1.03 meters.

Remember to take these results with a grain of humor. After all, you can't always trust everything a clown bot tells you!

To find the height of the Rizal monument, we can use trigonometry. Let's consider the triangle formed by the observer, the top of the monument, and the bottom of the monument.

First, let's convert the angles from degrees and minutes to decimal form.
The angle of elevation, 6° 50', can be converted to 6 + 50/60 = 6.8333°.
The angle of depression, 7° 30', can be converted to 7 + 30/60 = 7.5°.

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

Using the tangent function, we have:

tan(6.8333°) = h / x (Equation 1) - where x is the distance between the observer and the monument's base.

And,
tan(7.5°) = (h + 9) / x (Equation 2) - since the observer is standing 9m above the ground.

Rearranging Equation 1, we get:
h = x * tan(6.8333°)

Substituting this into Equation 2, we get:
tan(7.5°) = (x * tan(6.8333°) + 9) / x

Simplifying this equation, we have:
tan(7.5°) = tan(6.8333°) + 9/x

Now, let's solve for x by isolating it on one side of the equation:

tan(7.5°) - tan(6.8333°) = 9/x

Let's denote the expression on the left-hand side as 'A':

A = tan(7.5°) - tan(6.8333°)

Now, we can rewrite the equation as:
A = 9/x

To solve for x, we can rearrange the equation:

x = 9/A

Substitute the value of A into the equation and calculate the value of x.

Once we have the value of x, we can substitute it into Equation 1 to find the height of the monument, h = x * tan(6.8333°).

To find the height of the Rizal monument, we can make use of the properties of trigonometry.

Let's consider the triangle formed by the man, the top of the monument, and the bottom of the monument. We can label the height of the monument as 'h' and the distance between the man and the monument as 'x'.

Using the angle of elevation of 6° 50', we can determine the relationship between 'h', 'x', and the distance from the man to the bottom of the monument. Since the angle of elevation is the angle between the horizontal line and the line of sight from the observer to the top of the monument, we can form a right triangle with the observer, the top of the monument, and the horizontal line.

In this triangle, the opposite side is the height of the monument (h), the adjacent side is the distance between the observer and the monument (x), and the angle is 6° 50'. Applying trigonometry, we can use the tangent function:

tan(6° 50') = h / x

Now, let's consider the triangle formed by the man, the bottom of the monument, and the horizontal line. Using the angle of depression of 7° 30', we can establish the relationship between 'h', 'x', and the distance from the man to the bottom of the monument. The angle of depression is the angle between the horizontal line and the line of sight from the observer to the bottom of the monument.

In this triangle, the opposite side is the difference in height between the observer and the bottom of the monument (9m + h), the adjacent side is the same distance between the observer and the monument (x), and the angle is 7° 30'. Again, applying trigonometry and using the tangent function, we have:

tan(7° 30') = (9m + h) / x

Now, we have two equations that relate the unknown 'h' and the known 'x'. We can solve these equations simultaneously to find 'h', the height of the monument.

By rearranging the first equation, we get:

h = x * tan(6° 50')

Substituting this into the second equation, we have:

tan(7° 30') = (9m + x * tan(6° 50')) / x

Simplifying this equation, we will get:

tan(7° 30') = (9m / x) + tan(6° 50')

Now, we can solve this equation for 'x'. Rearranging the equation, we get:

(9m / x) = tan(7° 30') - tan(6° 50')

To find 'x', we need the values of tan(7° 30') and tan(6° 50'). We can use a scientific calculator or an online trigonometric calculator to obtain these values.

Once we have 'x', we can substitute it back into the equation h = x * tan(6° 50') to find the height 'h' of the Rizal monument.

Sketch the situation.

On my sketch, I labeled the top of the monument as A and its bottom as B, with B on ground level

I placed the man at a point 9 m above ground at some distance away from the monument and labeled that P
I drew a horizontal from P to meet AB at Q

I now have two right-angled triangles.
In triangle BPQ, BQ=9, angle BPQ = 7.5°
tan 7.5° = 9/PQ
PQ = 9/tan 7.5° = .....

in triangle APQ,
tan 6° 50' = AQ/PQ , but we know PQ

AQ = PQ(tan 6° 50') = ......

AB = 9 + AQ = .....