A person walks towards a tower. Initially when he starts, angle of elevation of the top of the tower is 30o . On traveling 20 metres towards the tower, the angle changes to 60o . How much more has he to travel to reach the tower? (1) 10 3 metres (2) 10 metres (3) 20 metres (4) 10 3 metres
Answers
10 meters distance is to be covered more by the person to reach the tower. (Option-2)
Given,
A person walks towards a tower.
Initially when he starts walking, the angle of elevation of the top of the tower = 30°
On traveling 20 metres towards the tower, the angle changes = 60°
To find,
Total distance to be covered more to reach the tower.
Solution,
We can simply solve this mathematical problem using the following process:
Let us assume that the the distance between the tower and the position of the person after traveling 20 metres towards the tower, is equal to x meters. Also, let us assume that the height of the tower is y meters.
Now, according to the question;
When the person initially starts walking, a right angled triangle is formed whose perpendicular is equal to the height of the tower and its base is the the distance between the tower and the position of the person. The line joining the person and the top of the tower firms the hypotenuse. the angle between the hypotenuse and base (the angle of elevation of the top of the tower for the person) is 30°.
Now, on applying Tan ratio for the given angle, we get;
Tan 30° = perpendicular/base
=> (height of the tower)/(distance between the tower and the position of the person) = 1/2
=> (height of the tower)/(the distance between the tower and the position of the person after traveling 20 metres towards the tower + 20 meters) = 1/√3
=> y/(x+20) = 1/√3
=> x + 20 = √3y
=> y = (x+20)/√3
{Equation-1}
Similarly, according to the question;
When the person has traveled 20 metres towards the tower, a right angled triangle is formed whose perpendicular is equal to the height of the tower and its base is the the distance between the tower and the position of the person. The line joining the person and the top of the tower firms the hypotenuse. the angle between the hypotenuse and base (the angle of elevation of the top of the tower for the person) is 60°.
Now, on applying Tan ratio for the given angle, we get;
Tan 60° = perpendicular/base
=> (height of the tower)/(distance between the tower and the position of the person) = 1/2
=> (height of the tower)/(the distance between the tower and the position of the person after traveling 20 metres towards the tower)= √3
=> y/x = √3
=> y = √3x
{Equation-2}
Now, according to equation-1 and equation-2;
(x+20)/√3 = √3x
=> x + 20 = 3x
=> 2x = 20
=> x = 10
=> the distance between the tower and the position of the person after traveling 20 metres towards the tower = 10 meters
Hence, 10 meters distance is to be covered more by the person to reach the tower. (Option-2)