From the top of the waterfalls, the angles of depression of the foot and the top of the tree are 39° and 24° , respectively. If the tree is 3 m from the foot of the waterfalls, how high is the waterfalls??
Did you make a sketch ?
I filled in all your data , and you actually have redundant information
we can simply take
tan 39° = h/3
h = 3tan39 = 2.44 m
34
To find the height of the waterfalls, we can use trigonometry. Let's assume that the height of the waterfalls is represented by "h."
From the information provided, we know that the angle of depression from the top of the waterfalls to the foot of the tree is 39°. This means that the angle between the line of sight and the horizontal is also 39°.
Similarly, the angle of depression from the top of the waterfalls to the top of the tree is 24°. This means that the angle between the line of sight and the horizontal is also 24°.
Now, let's draw a right-angled triangle to represent the situation, with the waterfalls at the top and the tree at the bottom. The distance between the foot of the tree and the waterfalls is 3 meters.
Using the tangent function, we can set up two equations:
tan(39°) = h/3 (equation 1)
tan(24°) = h/x (equation 2)
Solving equation 1 for h:
h = 3 * tan(39°)
Calculating h:
h ≈ 3 * 0.809784 = 2.429352 meters
Therefore, the height of the waterfalls is approximately 2.43 meters.
To find the height of the waterfall, we can use trigonometry and apply the concept of angles of depression.
Let's denote the height of the waterfall as 'h' (in meters).
And let's denote the distance from the top of the tree to the foot of the waterfall as 'd' (in meters), which is given as 3m in this case.
From the given information, we know that the angle of depression from the top of the waterfall to the foot of the tree is 39°.
In trigonometry, when we have an angle of depression, it forms a right triangle with the horizontal line, the line of sight, and the line between the object being observed and the observer.
Using the angle of depression, we can determine the vertical component of the triangle, which is the height of the waterfall (h).
Now, let's break down the problem into two right triangles:
1. The top view triangle: This triangle will have the height of the waterfall (h) as the vertical side, the horizontal distance from the top of the tree to the foot of the waterfall (d) as the base, and the angle of depression as 24°.
2. The side view triangle: This triangle will have the height of the waterfall (h) as the vertical side, the horizontal distance between the top of the tree and the observer's eye as the base, and the angle of depression as 39°.
Now, let's use trigonometry to calculate the height of the waterfall (h) using each triangle:
1. Using the top view triangle:
In this triangle, we have the angle of depression (24°) and the base (d = 3m).
We can use the tangent function to find the height:
tangent(24°) = height (h) / base (d)
tan(24°) = h / 3m
2. Using the side view triangle:
In this triangle, we have the angle of depression (39°) and the base (d = 3m).
We can again use the tangent function to find the height:
tangent(39°) = height (h) / base (d)
tan(39°) = h / 3m
Now, we have two equations with two unknowns. We can solve them simultaneously to find the value of 'h', the height of the waterfall.