a tree is on one side of a river two point p and q 15metre apart are chosen on the other side of the river in a straight line with the foot of the tree on a horizontal plane.the angle of elevation of the top of the tree from p and q 32degree and 51degree respectively 1 find the lenght of the tree 2distance of the tree point q
The answer is 13.6246
Please help to solve these question i want to know it
To solve this problem, we'll use basic trigonometry concepts.
1. Finding the length of the tree:
Let's assume the height of the tree is h.
Using the angle of elevation from point P, we can set up the following equation:
tan(32°) = h / x (Equation 1)
where x is the distance from the foot of the tree to point P.
Similarly, using the angle of elevation from point Q, we have:
tan(51°) = h / (x + 15) (Equation 2)
where (x + 15) is the distance from the foot of the tree to point Q.
Now, we can solve these two equations to find the value of h:
From Equation 1: x = h / tan(32°) (Equation 3)
From Equation 2: tan(51°) = h / (h / tan(32°) + 15)
Simplifying this equation:
tan(51°) = h / (h / tan(32°) + 15) = h * tan(32°) / (h + 15 * tan(32°))
Rearranging the equation:
h * (tan(51°) - tan(32°)) = 15 * tan(32°) * tan(51°)
h = (15 * tan(32°) * tan(51°)) / (tan(51°) - tan(32°))
2. Distance of the tree from point Q:
The distance of the tree from point Q is x + 15, as mentioned earlier.
Substituting the value of x from Equation 3, we get:
Distance from Q = h / tan(32°) + 15
Now, using the given values of the angles, we can calculate the final results:
h = (15 * tan(32°) * tan(51°)) / (tan(51°) - tan(32°))
Distance from Q = h / tan(32°) + 15
Solve these equations using a scientific calculator or any trigonometric calculator for the exact numerical values.
To find the length of the tree and the distance from the tree to point Q, we can use trigonometry and some basic geometry principles. Let's solve the problem step by step:
1. Finding the length of the tree:
Let's assume the foot of the tree is point T, and the top of the tree is point X. We need to find the length of TX.
We have two triangles formed: triangle TPX and triangle TQX. Both these triangles have a common side, TX. Let's analyze these triangles individually.
In triangle TPX:
- The angle of elevation of the top of the tree from point P is 32 degrees.
- Since TP is the horizontal plane, the angle between the horizontal plane TP and the line TX is 90 degrees.
In triangle TQX:
- The angle of elevation of the top of the tree from point Q is 51 degrees.
- Again, TQ is the horizontal plane, so the angle between the horizontal plane TQ and the line TX is 90 degrees.
Now, let's use the tangent function to find the lengths of TX for each triangle:
In triangle TPX:
tan(32 degrees) = TX / TP
In triangle TQX:
tan(51 degrees) = TX / TQ
We can rearrange these formulas to find TX:
TX = TP * tan(32 degrees) (Equation 1)
TX = TQ * tan(51 degrees) (Equation 2)
Given that TP and TQ are 15 meters apart, with TP + TQ = 15, we can substitute TQ = 15 - TP into Equation 2:
TX = (15 - TP) * tan(51 degrees) (Equation 3)
We now have two equations (Equation 1 and Equation 3) with two variables (TX and TP). We can solve this system of equations to find the value of TX, which will give us the length of the tree.
2. Finding the distance from the tree to point Q:
Once we find the length of the tree (TX), the distance from the tree to point Q (TQ) can be calculated as TQ = 15 - TP, as mentioned earlier.
Solving these equations will give us both the length of the tree (TX) and the distance from the tree to point Q (TQ).