Question

from the top of a 120m high tower , a man observes two cars on the opposite sides of the tower and in straight line with the base of tower with angles of depression as 60° and 45°.find the distance between the two cars.(take√3=1.732)

Answer 1

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Answer 2

Given,

Height of a tower = 120 meters

Angles of depression for the two cars on either side of the tower = 60° and 45°

To find,

The total distance between the two cars.

Solution,

We can simply solve this mathematical problem using the following process:

Let us assume that the distance between the base of the tower and the first car is x meters. And, the distance between the base of the tower and the second car is y meters.

Mathematically,

In a right-angled triangle, if a base angle (other than the right angle) is A, then,

Tan A = (the side opposite to A)/(side adjacent to A, other than the hypotenuse)

Now, according to the question;

The top of the tower, the base of the tower, and the first car forms a right-angled triangle, with the side joining the car and the top of the tower as the hypotenuse. The angle between the hypotenuse and the tower is 60°.

Now,

Tan 60° = (distance between the first car and base of the tower)/(height of the tower)

=> √3 = x/120 meters

=> x = 120√3 meters

=> x = 207.84 meters

And, the top of the tower, the base of the tower, and the second car form a right-angled triangle, with the side joining the car and the top of the tower as the hypotenuse. The angle between the hypotenuse and the tower is 45°.

Now,

Tan 45° = (distance between the second car and base of the tower)/(height of the tower)

=> 1 = y/120 meters

=> y = 120 meters

=> y = 120 meters

So, the total distance between the two cars

= (the distance between the base of the tower and the first car) + (the distance between the base of the tower and the second car)

= (x+y) meters

= 207.84 meters + 120 meters

= 327.84 meters

Hence, the total distance between the two cars is equal to 327.84 meters.