Question

From the foot of a hill the angle of elevation of the tip of a tower is found to be 45°. After walking 2 km upwards along the slope of the hill ,which is inclined at 30°, the same is formed to be 60°. Find the height of the hill along with the tower

Answer 1

Answer:

Hill height ≈ 1577 m

Tower height ≈ 1155 m       (very unrealistic!)

Step-by-step explanation:

Let a, b, c, d, e be the lengths as indicated in the diagram.

So height of the hill is c+d and the height of the tower is e.

Then...

a = 2 cos 30° = √3

c = 2 sin 30° = 1

(d + e) / b = tan 60° = √3  =>  e + d = b√3    ... (1)

(c + d + e) / (a + b) = tan 45° = 1  =>  c + d + e = a + b     ... (2)

Subtracting (1) from (2):

c = a + b - b√3

=> 1 - √3 = b ( 1 - √3 )

=> b = 1

( c + d ) / ( a + b ) = tan 30° = 1 / √3

=> ( c + d )√3 = a + b

=> ( 1 + d )√3 = √3 + 1

=> d√3 = 1

=> d = 1 / √3

Finally, from (1):

e = b√3 - d = √3 - 1/√3 = 2/√3

Therefore, the height of the hill is

c+d = 1 + 1/√3 ≈ 1577 m

and the height of the tower is

e = 2/√3 ≈ 1155 m