When viewing Angel Falls (the world's highest waterfall) from Observation Platform A, located on the
same level as the bottom of the waterfall, we calculate the angle of elevation to the top of the waterfall to
be 69.30°. From Observation Platform B, which is located on the same level exactly 1000 feet from the
first observation point, we calculate the angle of elevation to the top of the waterfall to be 52.90°. How
high is the waterfall?
To find the height of the waterfall, we can use a trigonometric relationship involving the angles of elevation from different observation points.
Let's label the height of the waterfall as "h" and the distance between Observation Platform A and the top of the waterfall as "x".
From Observation Platform A, the angle of elevation to the top of the waterfall is 69.30°. This means that if we draw a right triangle with one side along the horizontal ground from Observation Platform A to the top of the waterfall (which has length x) and another side going upwards with a height of h, the angle between these two sides will be 90°, and the angle opposite to the height h will be 69.30°.
Now, let's consider Observation Platform B, which is located on the same level exactly 1000 feet from Observation Platform A. From Observation Platform B, the angle of elevation to the top of the waterfall is 52.90°. In this case, the height h remains the same as in the previous triangle, but the length of the base has increased to x + 1000 (due to the 1000 feet distance between the two observation points).
We can now create a new triangle for Observation Platform B, where we have a right angle (90°), an angle opposite to the height h (52.90°), and a base with a length of x + 1000.
Now, we have two right triangles with the same height h but different base lengths. To find the value of h, we can set up a trigonometric equation based on the tangent function.
From the first triangle (Observation Platform A), we have:
tan(69.30°) = h / x
From the second triangle (Observation Platform B), we have:
tan(52.90°) = h / (x + 1000)
By rearranging both equations, we can solve for h:
h = x * tan(69.30°)
h = (x + 1000) * tan(52.90°)
Since both expressions are equal to h, we can set them equal to each other and solve for x:
x * tan(69.30°) = (x + 1000) * tan(52.90°)
Now, we can solve this equation to find the value of x. Once we have that value, we can substitute it back into either of the original equations to find the height h of the waterfall.