 Hi and welcome to the session. Let us discuss the following question. Question says, 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 degrees, 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 degrees. Find the time taken by the car to reach the foot of the tower from this point. Here we are given that a man standing at the top of the tower observes a car at an angle of depression of 30 degrees. Now, six seconds later, the angle of depression of the car is found to be 60 degrees. Let us now start the solution. First of all, let us draw a simple diagram to represent the problem. Here, AB is the tower, CB is the straight highway leading to the foot of the tower, B is the foot of the tower. So, we can write, let AB represents the tower, CB is the straight highway leading to the foot of the tower, that is, B. Now we know, man is standing at the top of the tower and he is observing car at point C at an angle of depression of 30 degrees. We know angle of depression is the angle found by the line of sight with the horizontal when it is below the horizontal level. So, angle PAC is the angle of depression of point C from point A. Now, we can write angle PAC is equal to 30 degrees. Now, we know PA is parallel to CB and AC is transversal. So, angle PAC is equal to angle ACB. Really, we can see these two are alternate angles. So, we can write PA is parallel to CB and AC is transversal. So, angle PAC is equal to angle ACB is equal to 30 degrees. Now, after 6 seconds, car reaches at point T on the highway. Now, angle of depression changes to 60 degrees. So, angle PAD is equal to 60 degrees. Now, PA is parallel to CB and AB is transversal. So, angle PAD is equal to angle ADB as they are alternate angles. Now, we can write angle PAD is equal to 60 degrees. Now, we know PA is parallel to CB and AD is transversal. So, angle PAD is equal to angle ADB is equal to 60 degrees since they are alternate angles. Now, we have given that car has uniform speed. So, let us assume that speed of the car is x meters per second. Now, distance covered by the car in 6 seconds that is CD is equal to 6 multiplied by x or we can say 6x meters. You know, distance is equal to speed into time. So, speed is equal to x meters per second and time is 6 seconds. So, distance covered in 6 seconds is equal to 6x meters. Now, let us consider right triangle ABD. In right triangle ABD, we know tan 60 degrees is equal to AB upon BD. We know tan theta is equal to perpendicular upon base. Here, theta is equal to 60 degrees perpendicular is AB and BD is the base. Now, we know tan 60 degrees is equal to root 3. So, substituting root 3 for tan 60 degrees we get root 3 is equal to AB upon BD. Now, multiplying both the sides by BD, we get BD root 3 is equal to AB or we can simply write it as AB is equal to BD root 3. Let us name this expression as 1. Now, in right triangle ABC tan 30 degrees is equal to AB upon BC. We know tan theta is equal to perpendicular upon base. In triangle ABC, AB is perpendicular, CB is base and 30 degrees is theta. So, we know tan 30 degrees is equal to 1 upon root 3. So, substituting 1 upon root 3 for tan 30 degrees we get 1 upon root 3 is equal to AB upon BC. Now, multiplying both the sides by BC we get BC upon root 3 is equal to AB or we can write it as AB is equal to BC upon root 3. Now, we know BC is equal to CD plus BD. So, we can write AB is equal to CD plus BD upon root 3. We have shown here that CD is equal to 6 x meters. Now, substituting this value of CD in this expression we get AB is equal to 6 plus BD upon root 3. Now, let us name this expression as 2. From expression 1 we know AB is equal to BD root 3. Now, substituting value of AB from 1 into we get BD root 3 is equal to 6 x plus BD upon root 3. We have substituted BD root 3 for AB in this expression. Now, multiplying both the sides by root 3 we get 3 BD is equal to 6 x plus BD. Now, subtracting BD from both the sides we get 2 BD is equal to 6 x. Now, dividing both the sides by 2 we get BD is equal to 3 x. So, we get BD is equal to 3 x meters. Now, we have to find the time taken by the car to reach the foot of the tower from point D. Now, we know BD that is the total distance travelled by the car from point D to foot of the tower is equal to 3 x meters and speed of the car is uniform that is x meters per second. So, time taken by the car to reach the foot of the tower from point D is equal to 3 x divided by x. We know time is equal to distance upon speed distance covered by the car that is BD is equal to 3 x meters and speed of the car is equal to x meters per second. Now, x and x will cancel each other and we get required time is equal to 3 seconds. So, 3 seconds is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.