a tree is x meter high the angle of elevation of its top from a point p on the ground is 23 degree . from another point q 10 meter from p and in line with p and foot of the tree , the angle of elevation is 32 degree. find x
To find the height of the tree (x), we can use trigonometry. Let's break down the information given step by step:
1. Point P is on the ground, and the angle of elevation from P to the top of the tree is 23 degrees.
2. Point Q is 10 meters away from P and in line with P and the foot of the tree. The angle of elevation from Q to the top of the tree is 32 degrees.
Now, let's use trigonometry to find x:
First, let's consider the triangle formed by Point P, Point Q, and the foot of the tree:
In this triangle, we have:
- Angle Q = 32 degrees (given)
- Side PQ = 10 meters (given)
- Side PQ is opposite to angle Q
- Let's call the height of the tree HQ (we don't know it yet)
Using the tangent function, we can write the following equation:
tan(32°) = HQ / PQ
Next, let's consider the triangle formed by Point P, the top of the tree, and the foot of the tree:
In this triangle, we have:
- Angle P = 23 degrees (given)
- Side PQ = 10 meters (given)
- Side x is opposite to angle P
- Side HQ is adjacent to angle P
Using the tangent function, we can write the following equation:
tan(23°) = x / HQ
Now, we can solve these two equations together to find the height of the tree (x):
From the first equation: tan(32°) = HQ / 10
Therefore, HQ = 10 * tan(32°)
Substitute this value into the second equation:
tan(23°) = x / (10 * tan(32°))
Solving for x, we get:
x = 10 * tan(23°) / tan(32°)
Calculating this expression will give us the height of the tree (x).
make a sketch, label the top of the tree A, and its bottom as B
label the point with the 23° angle P and the point with the 32°angle Q
We can find all the angles in triange PQA
angle P = 23, angle PQA = 148 , then angle PAQ = 9°
by sine law:
AQ/sin23 = (p-q)/sin9
AQ = (p-q)sin23/sin9
in the right-angled triangle, assuming the tree is vertical
sin 32 = x/AQ
x = AQ sin32
x = ((p-q)sin23/sin9) sin32