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 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 time taken by the car to reach the foot of the tower.

Step 1: Define variables and set up trigonometric relations

Let $AD$ be the height of the tower, $CD$ be the distance of the car from the foot of the tower at the second observation, and $BD$ be the distance at the first observation. We use the tangent function to relate the angles of depression to the sides of the right-angled triangles formed. From triangle $ADC$, we express the height $AD$ in terms of $CD$. Similarly, from triangle $ADB$, we express $AD$ in terms of $BD$.

Step 2: Equate expressions for tower height

Since the height of the tower $AD$ is constant, we can equate the two expressions for $AD$ obtained in the previous step. This allows us to establish a relationship between $BD$ and $CD$. Multiplying both sides by $\sqrt{3}$ simplifies the equation to $3CD = BD$.

Step 3: Relate distances to speed and time

The distance covered by the car in 6 seconds is $BC$. We can express $BC$ as the difference between $BD$ and $CD$. Substituting the relationship $BD = 3CD$ into this equation, we find that $BC = 2CD$.

Step 4: Calculate time to cover distance BC

We are given that the car covers the distance $BC$ in 6 seconds. Since the car moves with a uniform speed, we can calculate its speed by dividing the distance $BC$ by the time taken. Substituting $BC = 2CD$, we find the speed of the car to be $CD/3$ meters per second.

Step 5: Calculate time to reach the foot of the tower

To find the time taken by the car to reach the foot of the tower from point $C$, we divide the distance $CD$ by the car's speed. Substituting the speed we calculated, we find that the time taken is 3 seconds.

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