Points A and B are on the same horizontal line with the foot of a hill and the angles of depression of these points from the top of the hill are 35° and 25°, respectively. If the distance between A and B is 75 m, what is the height of the hill?
a diagram and a review of the basic trig functions should convince you that the height h of the hill is found via
h cot25° - h cot35° = 75
h cot25° + h cot35° = 75
hmmm. It appears we disagree on whether A and B are on the same side of the hill. Since we're talking about a hill, and not a pole, I stick by my interpretation -- heck -- I even have a drawing to show it!
To find the height of the hill, we can use trigonometry and the concept of angles of depression.
Let's denote the height of the hill as 'h'. We can set up two right-angled triangles to represent the situations at points A and B.
At point A:
Let's consider the right-angled triangle formed by the hill, point A, and the top of the hill. The angle of depression at point A is given as 35°.
Using trigonometry, we can write:
tan(35°) = h / x
where x is the horizontal distance between point A and the foot of the hill.
At point B:
Similarly, considering the right-angled triangle formed by the hill, point B, and the top of the hill, the angle of depression at point B is given as 25°.
Using trigonometry again, we can write:
tan(25°) = h / (75 - x)
where (75 - x) is the horizontal distance between the foot of the hill and point B. (We subtract x from 75 because points A and B are on the same horizontal line.)
Now, we have two equations:
tan(35°) = h / x
tan(25°) = h / (75 - x)
To solve for h, we can eliminate x from the equations by rearranging the first equation to express x in terms of h:
x = h / tan(35°)
Substituting this value of x into the second equation, we can solve for h:
tan(25°) = h / (75 - h / tan(35°))
Simplifying this equation will give us the value of h, which represents the height of the hill.