from a bridge of a boat on the niagara river, the angle of elevation to the top of niagara falls is 60 degrees. the angle of depression to the bottom of the falls is 15 degrees. if the bridge of the boat is 4.5m above the water, how high is the niagara falls?
Let's denote the height of Niagara Falls as h.
From the bridge of the boat, the angle of elevation to the top of the falls is 60 degrees. This means that the height of Niagara Falls can be expressed as h = (4.5 + x)tan(60), where x is the distance of the boat from the base of the falls.
Similarly, the angle of depression to the bottom of the falls is 15 degrees. This means that the height of Niagara Falls can also be expressed as h = (4.5 - x)tan(15).
Setting these two expressions equal to each other, we get (4.5 + x)tan(60) = (4.5 - x)tan(15).
Simplifying the equation, we have:
(4.5 + x)tan(60) = (4.5 - x)tan(15)
Multiplying through by the common denominator, tan(60)tan(15), we get:
(4.5 + x)tan(60)tan(15) = (4.5 - x)tan(15)tan(60)
We can now solve for x:
(4.5 + x)tan(60)tan(15) = (4.5 - x)tan(15)tan(60)
Dividing both sides by tan(15)tan(60), we have:
4.5 + x = 4.5 - x
2x = 0
x = 0
Since x represents the distance of the boat from the base of the falls and the boat is on the bridge, we can conclude that the boat is exactly at the base of the falls.
Therefore, the height of Niagara Falls is h = (4.5 + 0)tan(60) = 4.5tan(60) = 4.5 * √3 ≈ 7.8 m.
So, the height of Niagara Falls is approximately 7.8 meters.