Q) A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30∘, which is approaching to the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60∘. Find the further time taken by the car to reach the foot of the tower.

Ans:

Step 1: Diagram for this question:

Here, let’s consider AB is the tower and the car initially at point D, and then after 6 seconds, it is at point C.

Given that the angle of depression from point B at the Tower top to the car at Point D is 30⁰, the elevation angle from D to point B will be 30⁰.

Similarly, since the angle of depression from point B at the Tower top to the car at Point C is 60⁰, the elevation angle from C to point B will be 60⁰.

Let’s consider the distance covered by car from point D to point C is D₁ and distance from C to point A at foot of the tower is D₂.

Step 2:Find the distance D₁:

In Δ ABD,

….. (i)

Similarly, in Δ ABC,

….. (ii)

We can see that CD = AD – CD or D₁ = AD – D₂

Substituting the values from equations (i) and (ii), we get

Step 3: Find the speed of car

Now, it is given that this distance CD is covered in 6 secs

We know that:

Step 4: Find the time taken to cover CA

Now the car will cover distance with the speed of

We know that:

Substituting the values of distance and speed, we get

Therefore, the distance from point C to point A will be covered in 3 secs.

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