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.
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Step-by-Step Solution
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 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.