2. A person walking along a straight road observed that at two consecutive kilometre stones the angles of elevation of a hill front of him are 40o and 80o respectively. Find the height of the hill.
Make a sketch
I have A, B, and C on a horizontal line, and H as the top of the hill
Angle A = 40° and angle HBC - 80°, AB = 2 km
In triangle ABH, angle ABH = 100°
then angle AHB = 40°
which makes BH = 2 , (isosceles triangle)
(that was lucky, else I would need the sine law to find BH)
in triangle HBC (right-angled)
sin 80 = HC/2
HC = 2sin80 = 1.97 km
Wow, that is more than just a "hill"
To find the height of the hill, we can use trigonometry and the concept of angles of elevation.
Let's assume that the height of the hill is 'h' and the horizontal distance between the person and the hill is 'x'.
From the given information, we have two right-angled triangles:
In the first triangle, the angle of elevation is 40 degrees. The side opposite to this angle is 'h' (height) and the side adjacent to this angle is 'x' (horizontal distance).
In the second triangle, the angle of elevation is 80 degrees. The side opposite to this angle is also 'h' (height) and the side adjacent to this angle is 'x + 1' (horizontal distance).
Using trigonometry, we can use the tangent function to relate the angles and sides of the triangle:
In the first triangle:
tan(40°) = h / x
In the second triangle:
tan(80°) = h / (x + 1)
Now, we can solve these two equations simultaneously to find the values of 'h' and 'x'.
Dividing the two equations:
tan(40°) / tan(80°) = (h / x) / (h / (x + 1))
tan(40°) / tan(80°) = x + 1 / x
Using the values of tangent:
0.8391 = x + 1 / x
Multiplying through by 'x':
0.8391x = x + 1
0.8391x - x = 1
0.8391x = x + 1
Simplifying further:
0.1609x = 1
x = 1 / 0.1609
x ≈ 6.20
Now, we can substitute this value of 'x' back into one of the initial equations to find the height 'h'.
Using the first equation:
tan(40°) = h / x
tan(40°) = h / 6.20
h = tan(40°) * 6.20
Calculating this using a scientific calculator, we find:
h ≈ 5.03 meters
Therefore, the height of the hill is approximately 5.03 meters.