Points A and B are on the same horizontal line from the foot of a hill and the angles of depression of these points from the top of the hill are 30deg and 22deg, respectively. If the distance between A and B is 75m, what is the height of the hill?
The 2 lines of sight represent the
hypotenuse of 2 rt. triangles, and
the height of the hill represents the vertical side of both triangles. The
hor. side of smaller triangle(d) is
measured from the bottom of hill to A.
The hor. side of the larger triangle
is = to d + 75m.
Tan30 = h/d, h = d * Tan30.
h = (d+ 75)* Tan22
The 2 lines of sight represent the
hypotenuse of 2 rt. triangles, and
the height of the hill represents the vertical side of both triangles. The
hor. side of smaller triangle(d) is
measured from the bottom of hill to A.
The hor. side of the larger triangle
is = to d + 75m.
Tan30 = h/d,
h = d * Tan30.
h = (d + 75)* Tan22
Substitute d * Tan30 for h.
d * Tan30 = (d + 75) * Tan22.
Solve for d:
0.5774d = (0.4040d + 30.30.
0.5774d - 0.4040d = 30.30
0.1774d = 30.30.
d = 30.30/0.1774 = 174.74m.
h = 174.74 * Tan30 = 100.9m = height
of hill.
27
To find the height of the hill, we can use trigonometry. We have two angles of depression, 30° and 22°, and we know the horizontal distance between points A and B is 75 meters.
Let's denote the height of the hill as "h".
First, let's consider the angle of depression of 30°. We can draw a right triangle with the height of the hill, the horizontal distance between A and B, and the line of sight from the top of the hill to point A.
Using trigonometric ratios, we know that the tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the hill (h), and the adjacent side is the horizontal distance (75 meters).
So, for the 30° angle of depression, we can write:
tan(30°) = h / 75
Similarly, for the 22° angle of depression, we can draw another right triangle with the same height of the hill, the horizontal distance between A and B, and the line of sight from the top of the hill to point B.
We can write the equation:
tan(22°) = h / 75
Now, we have two equations with two unknowns (h and the distance between A and B). We can solve these equations simultaneously to find the height of the hill.
Let's solve the equations:
From the first equation, we can isolate h by multiplying both sides by 75:
h = 75 * tan(30°)
Similarly, from the second equation:
h = 75 * tan(22°)
Calculating the values:
h ≈ 43.3 meters (rounded to one decimal place)
Therefore, the height of the hill is approximately 43.3 meters.