A man observes that the angle of elevation on the top of a tree is 24°,he walks 43 m towards the tree and observes that the angle of elevation is 37°,find the height of the tree?
I don't know it
angles in left triangle are 24 + (180-37) + A = 180
where A is up top
so 24 +A -37 = 0
A = 13 degrees
then law of sines in that triangle
sin 24 / z = sin 13 / 43 where z is the hypotenuse up in the little triangle
.407 / z = .225/43
z = 77.8
now the right triangle on the right by the pole
we know the hypotenuse is 77.8
we know the angle opposit the pole is 37 deg
so
sin 37 = h/77.8
h = 46.8 meters height
Make a sketch, label the origianal position A and the new position B so that
AB = 43
Label the top of the tree as P and its bottom Q
In triangle ABP you have angle A = 24°, angle ABP = 143°, so angle APB = 13°
by the sine law:
BP/sin24 = 43/sin13, BP = 77.75
In the right-angled triangle, sin37° = h/BP
h = BPsin37 = 46.79
Another type of solution is to use the cotangent, often used for this style
of question by oobleck, and results in
height = 43/(cot24 - cot37) to obtain the same result.
To find the height of the tree, we can use trigonometry. Let's break down the problem step by step:
1. Draw a diagram: Draw a horizontal line to represent the ground. Mark a point on it to represent the observer. From that point, draw a vertical line to represent the tree.
2. Label the known quantities: Let's label the distance between the observer's initial position and the tree as "x" (in meters), and the height of the tree as "h" (in meters).
3. Identify the relevant angles: The angle of elevation is the angle between the horizontal line and the line of sight from the observer to the top of the tree.
4. Set up trigonometric equations: We can use the tangent function, as it relates the opposite and adjacent sides of a right triangle. Thus, we have two equations:
- tan(24°) = h / x (for the initial position)
- tan(37°) = h / (x - 43) (after moving towards the tree)
5. Solve the equations:
- From the first equation, we can express h as h = x * tan(24°).
- Substitute this expression for h into the second equation:
tan(37°) = (x * tan(24°)) / (x - 43)
Rearrange the equation and solve for x:
tan(37°) * (x - 43) = x * tan(24°)
x * (tan(37°) - tan(24°)) = 43 * tan(37°)
x = (43 * tan(37°)) / (tan(37°) - tan(24°))
6. Calculate the height: Now that we have the value of x, we can substitute it back into the first equation to find the height:
h = x * tan(24°)
Using a scientific calculator, evaluate the numerator and denominator separately, then divide to find the value of x. Substitute this value back into the first equation to find the height of the tree.