the angle of depression of the top and the foot of a flagpole as seen from the top of a building145 metres away are 26 degree and 34 degree respectively .find the heights of pole and the building?
Make your sketch.
I labelled the top of the building P and its bottom Q, the top of the pole A and its bottom B.
So QA = 145 m
Enter all the given angles, after that all angles at A and B can be found
In triangle PQB , angle PBQ = 34° and
tan 34 = PQ/145
PQ = ..... <----- the height of the building
Using cos 34 = 145/PB , you can find PB
In triangle ABP, you know all the angles, and the side PB
Use the sine law to find AB
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h1 = ht. of flagpole.
h2 = distance from top of flagpole to top of bldg.
(h1+h2) = ht. of bldg.
Tan34 = (h1+h2)/145.
(h1+h2) =145*Tan34 = 98 m. = ht. of bldg.
Tan26 = h2/145.
h2 = 145*Tan26 = 71 m.
h1 = (h1+h2) - h2 = 98 - 71 = 27 m. = ht. of flagpole.
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Well, well, well, looks like we have a little geometry problem here. Time to clown around and solve it with a smile!
Let's call the height of the flagpole "h" and the height of the building "B". The distance from the top of the building to the top of the flagpole is "x".
Now, using some good old trigonometry, we can say that:
tan(26°) = h / (145 + x)
and
tan(34°) = B / (145 + x)
But hey, aren't we forgetting something important here? Oh right, the angle of depression and the angle of elevation are complementary angles! So, we can say:
tan(34°) = h / x
And since tangents are just ratios, we can rewrite our equations as:
h = x * tan(34°)
and
B = (145 + x) * tan(26°)
Now we have a nice little system of equations. Let's solve it and bring out the answer with style!
h = x * tan(34°)
h = x * 0.6947
h ≈ 0.6947x
B = (145 + x) * tan(26°)
B = (145 + x) * 0.4877
B ≈ 0.4877x + 70.7796
Oh, and let's not forget the best part! We can equate "h" and "B" since they're both equal to "x" after all. So:
0.6947x ≈ 0.4877x + 70.7796
0.6947x - 0.4877x ≈ 70.7796
0.207x ≈ 70.7796
x ≈ 341.9014
Well, well, the distance "x" is approximately 341.9014 meters. And since we know "h" = "x" * 0.6947, we can find the height of the flagpole:
h ≈ 341.9014 * 0.6947
h ≈ 237.8649
The height of the flagpole is approximately 237.8649 meters.
But wait, there's more! We can also find the height of the building "B" using the equation B ≈ 0.4877x + 70.7796:
B ≈ 0.4877 * 341.9014 + 70.7796
B ≈ 313.2426 + 70.7796
B ≈ 384.0222
So, the height of the building is approximately 384.0222 meters.
There you have it! The height of the flagpole is approximately 237.8649 meters and the height of the building is approximately 384.0222 meters. I hope I brought a smile to your face with this math circus act!
To find the height of the pole and the building, we'll need to use trigonometry and make use of the concept of angles of depression.
Let's label the angles: A is the angle of depression from the top of the building to the foot of the flagpole, and B is the angle of depression from the top of the building to the top of the flagpole.
We can draw a right-angled triangle to represent the situation. Let the horizontal distance from the top of the building to the flagpole be x, the height of the flagpole be h, and the height of the building be H.
From the given information, we have the following equations:
1. tan(A) = h/x
2. tan(B) = (h + H)/x
We can use these equations to find the values of x, h, and H.
First, let's find x:
tan(A) = h/x
x = h / tan(A)
Next, let's substitute the value of x in the second equation:
tan(B) = (h + H)/ (h / tan(A))
tan(B) = (h + H) * tan(A) / h
Now, let's solve for H:
H = (tan(B) * h - h) / tan(A)
Finally, let's substitute the values of A, B, and H and solve for h and H:
h = x * tan(A)
H = (tan(B) * h - h) / tan(A)
Given that A = 34 degrees, B = 26 degrees, and x = 145 meters, we can calculate the values of h and H accordingly.
I'll perform the calculations for you.