Question 12 of 12advanced🔧 ApplyNumerical4 marks

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 from this point.

Correct Answer

Let AB be the tower of height h. Let the car be at point C when the angle of depression is 30° and at point D when the angle of depression is 60°. Let the distance BD be x and the distance BC be y.

In right triangle ABD, the angle of depression is 60°. tan 60° = AB / BD 3\sqrt{3} = h / x => h = x3\sqrt{3}

In right triangle ABC, the angle of depression is 30°. tan 30° = AB / BC 1/3\sqrt{3} = h / y => h = y / 3\sqrt{3}

Equating the values of h: x3\sqrt{3} = y / 3\sqrt{3} => y = 3x

The distance traveled by the car in 6 seconds is CD. CD = BC - BD = y - x = 3x - x = 2x.

Speed of the car = Distance / Time = 2x / 6 = x/3 units per second.

Time taken to reach the foot of the tower from point D (distance x): Time = Distance / Speed = x / (x/3) = 3 seconds.

Thus, the time taken by the car to reach the foot of the tower is 3 seconds.

Exercise: EXERCISE 9.1 | Q: 15 | (Page Page 10-11)
For More Understanding

Explanation

This problem utilizes the definitions of angle of depression and trigonometric ratios provided in the context. By creating two right-angled triangles for the two positions of the car, we establish a relationship between the tower's height and the car's distance from the tower. Since the car travels with uniform speed, the ratio of distance and time remains constant. We find that the distance covered in the first interval is twice the remaining distance, allowing us to calculate the remaining time without knowing the actual speed or height.

Solution Steps

  1. Step 1: Define variables for the tower's height (h) and the distances from the tower at 60° (x) and 30° (y).

  2. Step 2: Use tan 60° in triangle ABD to establish h = x3\sqrt{3}.

  3. Step 3: Use tan 30° in triangle ABC to establish h = y/3\sqrt{3}.

  4. Step 4: Equate the expressions for h to find the relationship y = 3x.

  5. Step 5: Calculate the distance traveled in 6 seconds as y - x = 2x.

  6. Step 6: Determine the speed as (2x)/6 = x/3.

  7. Step 7: Calculate the time to cover the remaining distance x as x/(x/3) = 3 seconds.

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