A tree is "x"meters high. The angle of elevation of its top from a point P on the ground is 23degrees. Form another point Q, 10meters from P and in line with P and the foot of the tree, the angle of elevation is 32degrees. Find "x".
(Please don't give me the answer only.. I also need the way so I can learn from it. Thankyou :])
Make a neat diagram.
Label the bottom of the tree R and its top T
You should have two right angled triangles, PTR and QTR
Label PQ = 10, let QR = x and RT = h
In triangle QRT, tan 32º=h/x
h = xtan32
in triangle PTR, tan 23º = h/(x+10)
so h = (x+10)tan23
equate the two equations, (both are h=...)
xtan32 = (x+10)tan23
xtan32 = xtan23 + 10tan23
xtan32 - xtan23 = 10tan23
x(tan32-tan23) = 10tan23
x = 10tan23/(tan32-tan23)
Now you could solve for x now, and sub that back into
h = xtan32
I got h = 13.24, let me know if you got the same answer.
To solve this problem, we can use the concept of trigonometry and set up two right-angled triangles.
Let's assume the height of the tree is "x" meters.
First, let's look at the triangle formed by the point P, the top of the tree, and the foot of the tree.
In this triangle, the angle of elevation is 23 degrees. The opposite side is the height of the tree (x meters), and the adjacent side is the distance from the point P to the foot of the tree.
Using the trigonometric function tangent (tan), we can write the equation:
tan(23 degrees) = opposite/adjacent
tan(23 degrees) = x/0
However, we cannot divide anything by zero, so we cannot directly solve for x using this equation.
Next, let's look at the triangle formed by the point Q, the top of the tree, and the foot of the tree.
In this triangle, the angle of elevation is 32 degrees. The opposite side is still the height of the tree (x meters), and the adjacent side is the distance from the point Q to the foot of the tree, which is given as 10 meters.
Using the trigonometric function tangent (tan), we can write the equation:
tan(32 degrees) = opposite/adjacent
tan(32 degrees) = x/10
Now we have an equation that relates the known quantities. We can solve for x by rearranging the equation:
x = 10 * tan(32 degrees)
Using a scientific calculator, calculate the value of tan(32 degrees) and multiply it by 10. The result will give you the value of x, which is the height of the tree.
To find the height of the tree (x), we can use trigonometry and create a right triangle with the tree height, the distance from Point P to the tree, and the angles of elevation given at points P and Q.
Let's break down the problem step by step:
Step 1: Draw a diagram
Start by drawing a diagram representing the problem. Draw a vertical line to depict the tree with the unknown height (x). Mark Point P on the ground, which is the base of the tree, and Point Q, which is 10 meters away from Point P along a straight line towards the tree's foot.
Step 2: Define the given information
We are given two angles of elevation:
- The angle of elevation from Point P to the top of the tree is 23 degrees.
- The angle of elevation from Point Q to the top of the tree is 32 degrees.
- The distance from Point Q to Point P is 10 meters.
Step 3: Identify the right triangles
We can create two right triangles. For both triangles, the vertical side represents the height of the tree (x). The horizontal side represents the distance from Point P to the tree (let's call it h).
In the first triangle (formed by Point P, the top of the tree, and the horizontal line from P), the angle opposite to the height of the tree is 23 degrees.
In the second triangle (formed by Point Q, the top of the tree, and the horizontal 10-meter line), the angle opposite to the height of the tree is 32 degrees.
Step 4: Apply trigonometry
For the first triangle:
We can use the tangent function since we have the opposite (height) and adjacent (distance from Point P to the tree) sides. The formula is:
tan(θ) = opposite / adjacent
In this case, θ is 23 degrees, the opposite side is x (the height), and the adjacent side is h (the distance from Point P to the tree).
So, we have:
tan(23) = x / h
For the second triangle:
We apply the same formula, but this time θ is 32 degrees, the opposite side is also x (the height), and the adjacent side is 10 meters (the distance from Point Q to Point P).
So, we have:
tan(32) = x / 10
Step 5: Solve the equations
To find x, we need to solve the two equations simultaneously.
From the first equation, we can write:
x = h * tan(23) (equation 1)
From the second equation, we can write:
x = 10 * tan(32) (equation 2)
Remember, h in equation 1 is the same as the horizontal distance from Point P to the tree.
Step 6: Substitute values and solve for x
To find x, we need to know the value of h. However, we don't have that information given directly in the problem. Therefore, we need additional information or equations to determine the value of h.
If you provide more details or equations from the problem, I can assist you further in finding the value of x.